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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 .

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.

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 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.

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.

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.

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.

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.

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 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.

This paper presents finite element solutions of a near-tip elastic field of a straight, nano-scale crack in a two-dimensional, linear elastic, whole plane subjected to mode-I crack-face loads. The mathematical model is formulated using a continuum-based theory of classical linear elasticity together with Gurtin-Murdoch surface model to capture the role of the residual surface tension present on the crack-face material layer. The formulation finally yields a second-order, integrodifferential equation governing the crack opening displacement. A weighted residual technique along with the regularization procedure is applied to establish a weakly singular weak-form equation with the involved kernel of O(ln )r . Galerkin strategy and the finite element procedure are then employed to discretize the weak-form equation. Various types of element shape functions, generated by standard C0 -elements, standard C1 -elements, and special elements with built-in crack-tip functions, are considered. A proper quadrature rule is selected to efficiently and accurately evaluate both nearly and weakly singular double line integrals over pairs of elements resulting from the discretization and the solution of a dense system of linear algebraic equations is obtained using an efficient indirect solver. The rate of convergence of finite element solutions is fully investigated and such information is then used to conclude the influence of the residual surface tension on the behavior of the near-tip field.

This paper aims to propose an analytical solution for dynamic analysis of the hetero junction carbon nanotubes (HJCNTs)-based mass nanosensors using a nonlocal strain gradient Timoshenko beam model. To have a more precise nanosensor, it is necessary to have deep information about the vibration characteristics of the nanostructure. So, two main goals are followed in this paper. At first, the vibration of HJCNTs with general (elastic) boundary conditions and without attached mass are studied using a proposed analytical solution. Afterward, the HJCNT is applied as a cantilever mass sensor for sensing light as well as heavy masses attached to its tip. For the large and heavy masses, the rotary inertia of the attached mass is also considered in the analysis. The governing differential equations are derived based on the Hamilton's principle and solved by an analytical method, which is based on the modified Fourier series. The weighted residual method is employed for obtaining the variationally consistent boundary conditions using the known equations of motion of the structure. The field quantities are obtained in the closed forms. The convergence and accuracy of the proposed solution are validated through some special cases available in the literature. The effects of small scale parameters and the elastic boundaries on the frequency and mode shapes of HJCNTs are studied. Moreover, the factors that affect the frequency shift of HJCNT-mass sensor are discussed. The obtained results introduce HJCNTs as new mass nanosensors that can operate more efficiently than uniform CNTs. This paper can be greatly useful in designing HJCNT-mass sensors and may serve as a benchmark for the future research in this field.

In this work, a three characteristic-lengths featured size-dependent gradient-beam is constructed by adopting the modified nonlocal model, resulting in much more general constitutive equation with stress gradient up to four-order and strain gradient to two-order. The six-order differential governing equation for transverse displacement is formulated. All boundary conditions especially variational consistent higher order boundary conditions of the present model are derived with the aid of weighted residual approach. The closed-form solutions to critical buckling loads under different sets of boundary conditions are systematically formulated with higher order boundary conditions incorporated. The numerical results show that both nonlocal parameters have significant effect on the buckling behaviors. Meanwhile, if two nonlocal parameters are taken as same, the present results cannot always reduce to that from Eringen's nonlocal model. Due to its clear physical meaning, the present model is expected to be widely adopted in mechanical analyses of nano-structures.

A highly efficient and accurate semi-analytical domain decomposition method for elastodynamic problems of functionally graded elliptic shells and panels with elastic constraints on the basis of the first-order shear deformation theory is presented. Firstly, according to the first-order shear deformation theory, the dynamic energy functional of elastic shell structure is established. Then, the multi-segment partitioning technique is used to segment the shell along the circumference and axis direction. Thirdly, the modified variational principle and least-squares weighted residual method are adopted to ensure the inherent continuity between segments. On this basis, the virtual springs are evenly arranged on each boundary to simulate the boundary forces, and then the desired boundary conditions can be simulated. The displacements of each shell domain are expanded as double Jacobi orthogonal polynomials in the circumferential and axial variable. Lastly, the piecewise matrices for a segment are assembled directly in a similar way to that of the finite element method, and the elastodynamic problems of functionally graded elliptic shells and panels are obtained by the variational operation with respect to generalized coordinate vectors. The numerical comparison shows high computational efficiency and accuracy of the present method. All the calculation results in this paper can be used as benchmark data for future scholars to study this structure.

In this paper a new enrichment technique is presented for weak singularities in isotropic and orthotropic 2D linear elasticity problems. With absolutely no information of the singularity order, in a novel numerical approach, the enriching bases are constructed through a weighted residual imposition of the partial differential equation. Conceptually, the bases can be categorized as equilibrated basis functions. As another advantage of the proposed technique, it will be shown that the enriched solution may be extracted for either isotropic or orthotropic materials in a similar manner, noting that singularity effect in orthotropic materials has not been as widely referred to as for isotropic materials in the literature. The presented examples selected from the well-known literature reveal the accuracy and applicability of the enrichment in the framework of a boundary type method. The method may also be found useful for the researchers focusing on the eXtended Finite Element Method or other similar mesh-based or mesh-less methods.

A numerical study is made on the entropy generation and magnetohydrodynamics natural convection of $$\hbox {Al}_{2}\hbox {O}_3$$ -water non-homogeneous nanofluid inside a square enclosure equipped with a heated trapezoidal body. The Galerkin weighted residual finite element method is applied to solve the dimensionless governing equations within the utilized computational domain along with the algorithm of Newton–Raphson iteration that is used for simplifying the nonlinear terms in the equations. The characteristics of fluid flow fields, temperature distributions and entropy generation are studied for an enormous range of the Rayleigh number $$(10^{3}\le Ra \le 10^{6})$$ , volume fraction of nanoparticles ( $$0\le \phi \le 0.04$$ ), Hartmann number $$(0\le Ha \le 50)$$ , thermal conductivity of the trapezoidal solid body ( $$k_{\mathrm{w}}=0.5$$ , 0.76, 1.95, 7 and 16) and the height of the trapezoidal solid body ( $$0.15 \le D \le 0.45$$ ). It is shown that the streamlines pattern is more sensitive to the increase in the Hartmann number in comparison with the augmentation of the volume fraction of nanoparticles. Also, for a more thermodynamically optimized system, the higher Hartmann number at a higher solid volume fraction of nanofluid is recommended as they show less entropy generation.

The inconsistences of the higher-order shear resultant expressed in terms of displacement(s) and the complete boundary value problems of structures modeled by the nonlocal strain gradient theory have not been well addressed. This paper develops a size-dependent Timoshenko beam model that considers both the nonlocal effect and strain gradient effect. The variationally consistent boundary conditions corresponding to the equations of motion of Timoshenko beams are reformulated with the aid of the weighted residual method. The complete boundary value problems of nonlocal strain gradient Timoshenko beams undergoing buckling are solved in closed forms. All the possible higher-order boundary conditions induced by the strain gradient are selectively suggested based on the fact that the buckling loads increase with the increasing aspect ratios of beams from the conventional mechanics point of view. Then, motivated by the expression for beams with simply-supported (SS) boundary conditions, some semiempirical formulae are obtained by curve fitting procedures.

An accurate frequency domain model is proposed to analyze the seismic response of uniform vertical cylinders with arbitrary cross section surrounded by water. According to the boundary conditions and using the variables separation method, the vertical modes of the hydrodynamic pressure are firstly obtained. Secondly, the three-dimensional wave equation can be simplified to a two-dimensional Helmholtz equation. Introducing the scaled boundary coordinate, a scaled boundary finite element (SBFE) equation which is a linear non-homogeneous second-order ordinary equation is derived by weighted residual method. The dynamic-stiffness matrix equation for the problem is furtherly derived. The continued fraction is acted as the solution of the dynamic-stiffness matrix for cylinder dynamic interaction of cylinder with infinite water. The coefficient matrices of the continued fraction are derived recursively from the SBFE equation of dynamic-stiffness. The accuracy of the present method is verified by comparing the hydrodynamic force on circular, elliptical and rectangle cylinders with the analytical or numerical solutions. Finally, the proposed model is used to analyze the natural frequency and seismic response of cylinders.

A FEM for unsaturated transient seepage is established by using a quadrilateral isoparametric element, considering the fact that the main permeability does not coincide with the axis situation. It creates a function by using the element’s node hydraulic head and shape function instead of the real head in the Richard seepage control equation. With the help of the Galerkin weighted residual method, a FEM equation is given for analyzing 2-dimensional transient seepage problem. Further, based on the Jacobi matrix and Gauss numerical integral, it determines the elements of stiffness and capacitance matrices. This FEM equation considers not only the anisotropic of soil but also the uncoincidence between permeability and the axis. It is a common form of transient seepage. In the end, two examples illustrate the node accuracy of the quadrilateral element and the correctness of this FEM equation.

In this paper, for studying the influence of the randomness of structural parameters of high-speed elevator lifting system (HELS) caused by manufacturing error and installation error, a continuous time-varying model of HELS was constructed, considering the compensation rope mass and the tension of the tensioning system. The Galerkin weighted residual method is employed to transform the partial differential equation with infinite degrees of freedom (DOF) into the ordinary differential equation. The five-order polynomial is used to fit the actual operation state curve of elevator, and input as operation parameters. The precise integration method of time-varying model of HELS is proposed. The determination part and the random part response expression of the longitudinal dynamic response of HELS are derived by the random perturbation method. Using the precise integration method, the sensitivity of random parameters is determined by solving the random part response expression of time-varying model of HELS, and the digital characteristics of the acceleration response are analyzed. It is found that the line density of the hoisting wire rope has the maximum sensitivity on longitudinal vibration velocity response, displacement response and acceleration response, and the sensitivity of the elastic modulus of the wire rope is smallest.

The harmonic balance method is a version of Galerkin’s weighted residual method. The harmonic balance is to seek an approximate periodic solution of a differential-algebraic equation by a truncated Fourier series. Substituting a guessed periodic solution into the differential equation leads to a residual term. As the approximate solution is periodic, so will be the residual term. Hence, it can also be expanded in a Fourier series. Harmonic balance then requires that the Fourier coefficients of the residual term vanish.

This chapter addresses computationally demanding numerical formulations in the biomechanics of soft tissues. The theory of mixtures can be used to represent soft hydrated tissues in the human musculoskeletal system as a two-phase continuum consisting of an incompressible solid phase (collagen and proteoglycan) and an incompressible fluid phase (interstitial water). We first consider the finite deformation of soft hydrated tissues in which the solid phase is represented as hyperelastic. A finite element formulation of the governing nonlinear biphasic equations is presented based on a mixed-penalty approach and derived using the weighted residual method. Fluid and solid phase deformation, velocity, and pressure are interpolated within each element, and the pressure variables within each element are eliminated at the element level. A system of nonlinear, first-order differential equations in the fluid and solid phase deformation and velocity is obtained. In order to solve these equations, the contributions of the hyperelastic solid phase are incrementally linearized, a finite difference rule is introduced for temporal discretization, and an iterative scheme is adopted to achieve equilibrium at the end of each time increment. We demonstrate the accuracy and adequacy of the procedure using a six-node, isoparametric axisymmetric element, and we present an example problem for which independent numerical solution is available. Next, we present an automated, adaptive environment for the simulation of soft tissue continua in which the finite element analysis is coupled with automatic mesh generation, error indicators, and projection methods. Mesh generation and updating, including both refinement and coarsening, for the two-dimensional examples examined in this study are performed using the finite quadtree approach. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. To allow the finite difference algorithm to proceed in time using an updated mesh, solution values must be transferred to the new nodal locations. This rezoning is accomplished using a projected field for the primary variables. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using a linear, two-dimensional, axisymmetric problem corresponding to the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients and to produce solutions in good agreement with independent, converged, numerical solutions.

Analytical solutions are presented for the torsion of bi-directional functionally graded (FG) linearly elastic truncated conical cylinders for six functional forms of the shear modulus varying in both the radial and the axial directions. Furthermore, for an arbitrary variation of the shear modulus along the two directions, we employ a weighted residual approach (WRA) to obtain accurate numerical solutions for the linear elastic boundary value problem. The influence of the variation of the shear modulus and of the cone angle on the stress distribution in the cylinder is discussed through numerical examples. The analytical solutions presented herein can serve as benchmarks for ascertaining the accuracy of approximate or numerical solutions. The WRA combined with an optimization algorithm can be used to find the shear modulus variation to maximize the torsional stiffness.

This study develops the extended Kantorovich method (EKM) to provide a closed form semi analytical solution for the bending analysis of two edge-bonded thin rectangular plates. The constituent plates could be different in thickness, length, material, loading conditions, and Winkler foundation’s stiffness. A combination of clamp, free, and simply supports are applied to the structure. The shared edge in the composite plate is assumed to be perfectly bonded. By applying the EKM together with the idea of weighted residual technique, two sets of ODEs are obtained. Bending is assumed to remain continuous on the bonded edge. The EKM procedure is modified by applying the coordinate of an arbitrary shared point in the boundary conditions for the shared edge, to relate the bending of the two plates. The ODEs are solved iteratively to obtain the deflection function in a fast convergence trend. Two examples of aluminium-steel plate and functionally graded material-steel plate are considered. The deflection results from the boundary modified EKM (BM-EKM) are in high agreement with the finite element solution results. The bending of stepped plates is a special case of the current study. The suggested BM-EKM strengthens the EKM’s ability for solving complex jointed/bonded structures in structural analyses.

We developed an algorithm to simulate magnetotelluric responses for three-dimensional general electrical anisotropic structures with finite element (FE) method. From the Maxwell's equations , the curl-curl equation in terms of the electric vector potential and the magnetic scalar potential is obtained by decomposing the magnetic field into the incident field and the resultant field. For this method, the only unknown variable in the air is the magnetic scalar potential. Galerkin weighted residual method is adopted to generate the variational equations. Then we made a comparison with the edge-based FE method which is in terms of electric fields to validate the accuracy of this method. Later, two types of anisotropic models are studied and the results are analyzed in detail. Finally, based on these analyses, two main meaningful conclusions are obtained.

In the current paper, lateral vibrations of embedded hetero-junction carbon nanotubes (HJCNTs) are investigated. HJCNT and its surrounding medium are idealized as two different connected nonlocal strain gradient Rayleigh beams and a two-parameter (Pasternak) elastic foundation, respectively. The governing differential equation is derived based on the Hamilton's principle, and the classical and higher order boundary conditions are obtained using the weighted residual approach. The sixth-order differential equation is then solved analytically and numerically. The differential quadrature element method based on the Hermit interpolation functions is employed and developed to the stepped nanobeams to establish the reliability of the analytical method. The effects of the nonlocal parameter, the strain gradient parameter, the type of boundary conditions, the type of adopted higher order boundary conditions and the elastic foundation parameters on the frequency and mode shapes are studied and discussed in details. This paper provides an appropriate source not only for the vibrational study of HJCNTs, but also for comparing the vibration results of embedded uniform or stepped nanobeams obtained based on different theories that consider the size-dependent effects.

The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.

In the present study, the detailed procedures of Galerkin weighted residual technique of finite element method (FEM) for solving two-dimensional incompressible natural convective flow of nanofluids using nonhomogeneous dynamic model are discussed for the first time. The physical domain is discretized by using unstructured triangular elements. The governing partial differential equations of nanofluids are made dimensionless using the suitable transformation of variables for weak formulations. The method of weighted residuals is used for obtaining the approximate solutions. This approach typically leads to a sparse and indefinite matrix that is difficult to solve efficiently. The formation of an indefinite matrix is avoided in the present work by introducing an artificial compressibility term in the continuity equation. Unequal order interpolation functions are used for pressure, velocity, temperature and concentration variables. The coefficient matrices are calculated using interpolation functions. Assembling of triangular elements in the discretized domain is discussed elaborately. The process of calculating boundary integrals is also discussed. The Newton-Raphson iteration technique along with Euler-backward scheme is used to solve the global matrix. The sample results are obtained for the convective flow of nanofluids in a concentric annulus. It shows that the annulus of having higher thickness is the best performer enhancing convective heat transfer rates.

It is important to understand how magnetotelluric (MT) modeling can most effectively be performed in general anisotropic media. However, previous studies in this area have mainly focused on the use of one-dimensional (1D) and two-dimensional (2D) algorithms. Thus, building on earlier work, it is important to study the performance of three-dimensional (3D) modeling in arbitrary conductivity media; therefore, an edge-based finite element (FE) method has been developed for 3D MT modeling in arbitrary conductivity media. This approach is based on the initial derivation of a series of equivalent variational equations that are based on Maxwell equations, generated using the weighted residual method. Specific values were then obtained for coefficient matrixes of this edge-based FE method using hexahedral meshes, and the algorithm was verified by comparing its results with finite difference (FD) solutions generated using a 2D anisotropic model. Finally, the results of a 3D anisotropic model were analyzed detailed for three conditions; another 3D anisotropic model was designed and its results were compared with two isotropic models'.

A domain-boundary element method is developed for elastodynamics of graded beams.Differential equations are derived considering variations of physical properties.Static fundamental solutions are used in weighted residual statements.Integral equations are reduced to time-dependent ordinary differential equations.Parametric analyses are carried out for graded beams under various loadings. A new domain-boundary element method is developed for elastodynamic analysis of functionally graded Timoshenko beams. Three governing partial differential equations of motion are derived by considering through-the-thickness variations of the physical properties. Weighted-residual forms are imposed utilizing the static fundamental solutions. These forms are then reduced to three integral equations containing domain integrals with time derivatives of unknown functions. Through domain discretization and shape function approximation, integral equations are converted to a system of ordinary differential equations in time. Forced dynamic response is revealed by solving the system of equations via Houbolt method. Comparison of dynamic responses generated for homogeneous beams to those calculated through an analytical solution and finite difference method verify the developed procedures. Further parametric analyses are performed for functionally graded Timoshenko beams under step, harmonic, and impulsive loadings. The numerical results presented illustrate the influence of material inhomogeneity on time histories of deflection and stress. Domain-boundary element method is demonstrated to be an effective technique for elastodynamic analysis of functionally graded structures.

We study the stability of weakly viscoelastic film (Walter's B″) flowing down under gravity along a slippery inclined plane. The focus is to investigate the interaction of the bottom slip with the viscoelastic parameter as well as the influence of the other flow parameters on the stability of the flow. For the slippery substrate, we use the Navier-slip boundary condition at the solid–liquid interface. The dimensionless slip length β is first assumed to be small and its order is considered same as the order of the film aspect ratio ϵ=H/L, where H is the mean film thickness and L is a typical wavelength. To discuss the coupled effect of slip length β and viscoelastic parameter γ, we have used the classical Benney equation model (BEM) as well as the weighted residual method (WRM). For linear stability, the normal mode analysis is carried out to capture the instability threshold. The critical Reynolds numbers (Rec) are obtained for BEM and WRM separately for the system. We found that the first-order WRM is a better choice to capture the instability threshold in comparison with a first-order BEM when β is small. Another noteworthy result we obtain is that, in the absence of β, WRM and BEM yield the same expression for the critical Reynolds number. Further, we have extended the study for moderate values of β, that is, β of order unity and it is found that both BEM and WRM are able to capture the effects of β and γ. We derive the Orr–Sommerfeld (OS) type eigenvalue problem and an asymptotic analysis is performed for small perturbation wavenumbers, which gives an expression for the critical Reynolds number for the instability of very long perturbations. The critical Reynolds number obtained by the OS eigenvalue problem exactly matches with that obtained by BEM. Finally, we validate our analytical predictions by performing a direct numerical simulation of the WRM and good agreement between the results of the linear stability analysis, weighted residual model, and the numerical simulations is found.

Wave based method is an efficient numerical technique that can be used to solve the structural dynamic problems, especially the vibration of plates. The approach expands the dynamic field variables to a weighted sum of wave functions, and the contribution factors are determined by minimizing the errors on the boundary conditions. While the governing equation is satisfied a priori, the boundary conditions are satisfied in the integral sense, using preferably a weighted residual formulation. However, the weighted residual formulation is not always suitable for all kinds of boundary conditions. In plate bending problems, the plate edges can be modelled as clamped, simply supported, free, symmetric or (visco) elastically restrained against translation or/and rotation, depending on how they are restrained in reality. But the currently available weighted residual formulations are suitable either only for the non-elastic restraints (clamped, simply supported, free and symmetric) or only for the general (visco) elastic restraints with both translational and rotational elasticities. In this paper, a general formulation is proposed for all the aforementioned types of edge restraints. It is equivalent to the conventional one when the plate edges are non-elastically restrained, but performs better with the introduction of corner residuals for the plate with general elastically restrained edges. Apart from the ideal non-elastic and the general full-elastic restraints, this modified formulation also works well for the partial-elastic ones with only either translational or rotational elasticity. Numerical cases are studied on a rectangular plate for all the restraint types, where the proposed formulation is validated. The restraint types can be identical or different among the plate edges, and how to apply the formulation to the non-uniformed situation is also demonstrated. Throughout the considered cases, this paper shows the accuracy and efficiency of the wave based method by comparing with the finite element method in computing the forced vibration of the thin plate. With the modified weighted residual formulation, the method further shows its great convenience in changing the restraint types or stiffness (and damping) parameters of plate edges.

The size effects on the dynamic behaviors of rods are investigated within the framework of the nonlocal strain gradient elastic theory. The variationally consistent boundary conditions are derived by using the weighted residual method with respect to the known equation of motion of rods. Similar to the fourth-order differential equation of motion for classical Euler–Bernoulli beams, the variationally consistent boundary conditions of nonlocal strain gradient rods are clarified for the first time. The exact characteristic equations for determining the longitudinal frequencies are presented for four types of boundary value problems. To gain insight into the asymptotic behaviors of rods, the reduced boundary values problems are also analytically given. The numerical results show that both the softening effect and the stiffening effect can be captured by adjusting the two material length parameters. However, when the two material length parameters are the same, the present results cannot recover to the corresponding classical ones for rods with general boundary conditions, which are different from the previous findings reported in the literature.

The airborne electromagnetic (AEM) method has a high sampling rate and survey flexibility. However, traditional numerical modeling approaches must use high-resolution physical grids to guarantee modeling accuracy, especially for complex geological structures such as anisotropic earth. This can lead to huge computational costs. To solve this problem, we propose a spectral-element (SE) method for 3D AEM anisotropic modeling, which combines the advantages of spectral and finite-element methods. Thus, the SE method has accuracy as high as that of the spectral method and the ability to model complex geology inherited from the finite-element method. The SE method can improve the modeling accuracy within discrete grids and reduce the dependence of modeling results on the grids. This helps achieve high-accuracy anisotropic AEM modeling. We first introduced a rotating tensor of anisotropic conductivity to Maxwell’s equations and described the electrical field via SE basis functions based on GLL interpolation polynomials. We used the Galerkin weighted residual method to establish the linear equation system for the SE method, and we took a vertical magnetic dipole as the transmission source for our AEM modeling. We then applied fourth-order SE calculations with coarse physical grids to check the accuracy of our modeling results against a 1D semi-analytical solution for an anisotropic half-space model and verified the high accuracy of the SE. Moreover, we conducted AEM modeling for different anisotropic 3D abnormal bodies using two physical grid scales and three orders of SE to obtain the convergence conditions for different anisotropic abnormal bodies. Finally, we studied the identification of anisotropy for single anisotropic abnormal bodies, anisotropic surrounding rock, and single anisotropic abnormal body embedded in an anisotropic surrounding rock. This approach will play a key role in the inversion and interpretation of AEM data collected in regions with anisotropic geology.

This article presents a new numerical scheme to approximate the solution of one-dimensional telegraph equations. With the use of Laplace transform technique, a new form of trial function from the original equation is obtained. The unknown coefficients in the trial functions are determined using collocation method. The efficiency of the new scheme is demonstrated with examples and the approximations are in excellent agreement with the analytical solutions. This method produced better approximations than the ones produced with the standard weighted residual methods.

In this chapter the adomian decomposition method, variational iteration method, differential transformation method, homotopy perturbation method, homotopy analysis method, and weighted residual method are presented including example cases. From a mathematical viewpoint, the nanofluids can be modeled in two main ways: single- and two-phase methods. As noted in Chapter 1, Introduction to nanofluids, for the single-phase model, the nanoparticles and base fluid are considered as a single phase, which means the equivalent properties can be considered for it. But in the two-phase model the governing equations must be solved for the both nanoparticle and fluid phases at the same time. To find the best model, comparison of single-and two-phase modeling is considered here.

Multi-term extended Kantorovich method (MTEKM) is used to solve the bending problem of thin skew (parallelogram) functionally graded plate resting on the Winkler elastic foundation under uniformly distributed transverse load. Formulations are based on the classical plate theory (CPT) with the physical neutral surface is considered as the reference plane. Various configurations of clamped, simply supported and free edges are considered. By implementing the concept of Galerkin's weighted residual method, the fourth-order partial differential governing equation and boundary conditions are converted into two sets of ordinary differential equations (ODE), which are then solved numerically using “Chebfun” numerical computation package. Convergence and accuracy of MTEKM are investigated. Results obtained with MTEKM are compared to finite element method (FEM) solutions. FEM has been implemented using ANSYS software, in which the plate is modeled with shell elements, while the elastic foundation is modeled as a pair of contact/target elements. In addition, the effects of both the Winkler foundation stiffness and material power index have been investigated. Applying MTEKM in bending analysis of thin skew plates offered more accurate results than the single-term EKM but with the cost of more computation time but still provides simplicity and rapid convergence. It is found that MTEKM well suits the bending problem of skew FGM plates resting on elastic foundation. © 2020 Karabuk University

This chapter starts with the analytical description of thick plate members. Thick plates are plates where the contribution of the shear force on the deformations is considered. Based on the three basic equations of continuum mechanics, i.e., the kinematics relationship, the constitutive law, and the equilibrium equation, the partial differential equations, which describes the physical problem, is derived. The weighted residual method is then used to derive the principal finite element equation for thick plate elements. The chapter exemplarily treats a four-node bilinear quadrilateral (quad 4) bending element.

In this paper, nonlocal free vibration of axial rods embedded in elastic medium is examined by utilizing Love–Bishop rod theory with finite element method for the first time in the literature. Firstly, kinematic relations and dynamic equilibrium of this rod formulation are constituted. Equation of motion and boundary conditions are attained with the help of first variation of total potential of nano rod and solved using separation of variables. The frequency equations of four different nanorod types are obtained. On the other hand, a size–dependent finite element formulation is presented based on Weighted Residual Method. The nondimensional frequency parameters of nanorods are calculated by using analytical and finite element methods for different parameters such as mode number, nonlocal parameter, rod length, length–to–diameter ratio, finite element number and a detailed discussion is performed in the numerical results. In order to understand the characteristics of this rod theory, several comparative results that include nonlocal simple rod formulation are given as well.

Investigating the dynamic behaviour of circular plate resting on elastic foundations are very important in designing of structural systems. This study examines the free vibration analysis of circular plate resting on Winkler and Pasternak foundations. The governing nonlinear partial differential equation is transformed to Duffing equation based on von Karman geometric nonlinear principle and the nonlinear to linear frequency ratios are obtained while the linear natural frequencies are determined using Galerkin of weighted residual method. Also, the accuracy and reliability of the approximate solutions obtained are demonstrated by comparing the obtained results with available results reported in the literature. The analytical solutions obtained are used for examining the effect of elastic foundations on the dynamic behaviour of the circular plate. From the results, it is observed that, increasing elastic foundation parameter increases the natural frequency. As the nonlinear foundation increases, the nonlinear vibration frequency ratio decreases. The nonlinear Winkler foundation attenuates the amplitude of vibration of the circular plate. It is hoped that, the present study will contribute to the existing knowledge of classical theory of vibration.

In the present work, we modified the conventional "weighted residual method" to some nonlinear evolution equations and tried to obtain the approximate progressive wave solutions for these evolution equations. For the illustration of the method we studied the approximate progressive wave solutions for the perturbed KdV and the dissipative NLS equations. The results obtained here are in complete agreement with the solutions of inverse scattering method. The present solutions are even valid when the dissipative effects are considerably large. The results obtained are encouraging and the method can be used to study the cylindrical and spherical evolution equations. Publisher's Version

In this study, a lamination parameter based optimization algorithm is developed in a finite element framework using classical plate deformation theory. The optimization reveals lamination parameter distribution over the plate which are then converted into fiber directions, stacking sequences and layer thicknesses by means of a material library. The optimum fiber angles in each ply are used to find continuous fiber angle distributions a using weighted residual method for the first time. The method uses finite element based least-squares and continuity constraint (LSC) to find a manufacturable design without making any preliminary assumption such as splines, stream functions, etc., for the fiber angle distributions. Penalty parameter is used to enforce a maximum limiting value for the fiber curvatures. The proposed method is applied to design for the minimum compliance in-plane and out-of-plane problems, and significant improvements are obtained in comparison to the literature findings.

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.