Newmark Time Integration Scheme Research Articles

Nonlinear dynamic response of a sandwich plate with negative Poisson’s ratio honeycomb-core layer under low-velocity collision impact

In this paper, a nonlinear analytical model is presented for the low-velocity collision impact of a sandwich plate with auxetic honeycomb-core layer. The auxetic feature of the honeycomb material is realized by mathematically expressing the effective material coefficients in terms of material property and cellular geometric parameters through a homogenization method. The higher order shear deformation theory, the von Kármán nonlinearity theory, and a modified Hertz contact law which accounts for the contact pressure distribution and indentation effect, are employed to establish the kinematic relations. The Newmark time integration scheme in conjunction with the direct iterative method is utilized to establish a solution procedure for the nonlinear dynamic governing equation. The verification of the presented model with the data in published literatures is carried out, followed by a series of numerical analyses for influences of cellular geometric features and impactor’s initial conditions (such as impactor’s initial velocity, mass, and nose curvature radius) on the nonlinear dynamic response. The results of numerical analyses show that the geometric features of the unit cell in the honeycomb-core and impactor’s initial conditions can cause significant influences on the nonlinear collision impact behaviors of the system.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

On the selected problems of time-dependent dynamics of composite truncated conical shells-like aerospace structures

In this paper, the nonlinear dynamic responses of ring-stiffened truncated conical sandwich shells-like aerospace structures fabricated by perfectly bonded graphene platelets (GPLs) reinforced composite porous core and face sheets (GPLRC-PC-FS) under asymmetric internal ring-shaped moving load are investigated. The sandwich shells have edge point supports, which are simulated using the translational and rotational artificial springs. In addition to the open cell porosities, closed cells are also considered and analyzed. The spatially discretized nonlinear motion equations are derived based on the first-order shear deformation theory (FSDT) under the von Kármán geometric nonlinearity assumptions using the Ritz method with Chebyshev polynomials as its admissible basis functions. Then, Newmark's time integration scheme along with the Newton–Raphson method is employed to obtain the shell dynamic responses. After demonstrating the robustness and accuracy of the approach, the influence of rings number, their size and arrangement, core porosity distribution and structure, face sheets, GPLs parameters, moving load velocity and boundary conditions on the shell responses are studied. The results show that the shell dynamic behaviors are significantly affected by porosity, but almost are independent of the type of porosity distribution pattern and its structure (open and closed cell porosities). It is shown that the shell deflection can be minimized by suitably choosing the number of rings, their dimensions and arrangement. Moreover, the nonlinear behaviors of the shells have a significant correlation with the number of point supports and by adjusting the support stiffnesses, the nonlinear dynamic responses of the shells can be reduced.

Transient analysis of functionally graded graphene oxide powders-reinforced porous composite beams resting on elastic foundations using the reproducing kernel particle meshless method

In this paper, we propose a numerical algorithm, combining the reproducing kernel particle collocation (RKPC) meshless method with the implicit Newmark time integration scheme to examine the transient response of porous composite beams (PCB) strengthened with functionally graded (FG) graphene oxide powders (GOP), and resting on elastic foundations (EF). The PCB are subjected to dynamic loads (DL) with different boundary conditions (BC), and rested on EF modeled using the Winkler–Pasternak foundation model. The first-order shear deformation theory (FSDT) is employed to model the studied PCB reinforced with FG-GOP. The Halpin–Tsai homogenization technique and the rule of mixture are used to determine the effective material properties of the PCB taking the porosity into consideration. The obtained equations of motion using the principle of virtual displacement, are rewritten in a matrix–vector strong form and solved using the adopted numerical strategy. The performance and efficiency of the present approach are validated by comparison with numerical results of the finite element method (FEM). Parametric studies are performed to investigate the effect of GOP weight fraction, GOP distribution patterns, boundary conditions, porosity volume fraction, size of GOP, and elastic foundation stiffness on the transient behavior of FG-GOP reinforced PCB.

Nonlinear dynamic buckling and vibration of thermally post-buckled temperature-dependent FG porous nanobeams based on the nonlocal theory

In this paper, nonlinear dynamic snap-through buckling and vibration behavior of the thermally post-buckled functionally graded (FG) porous nanobeams subjected to static and sudden mechanical loads are investigated utilizing the nonlocal elasticity theory. The physical properties of the nanobeam are considered to be functions of temperature based on the Touloukian model. In addition, to describe the FG porous materials, two different patterns of porosity distribution are adopted using trigonometric functions through the thickness of the nanobeam. The equations of motion in conjunction with the von-Kármán nonlinear assumption are established in the framework of Hamilton’s principle. By employing the Chebyshev-Ritz procedure, the nonlinear equations are discretized for three types of edge supports. Following that, the cylindrical arc-length technique is employed to assess the vibrational responses of the post-buckled nanobeam during static snap-through buckling. To evaluate the nonlinear dynamic buckling of the graded nanobeam under a sudden dynamic load, the Newmark time integration scheme together with the Newton-Raphson iterative method are utilized. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. After validating the developed mathematical model, a comprehensive investigation is carried out to determine the role of various physical and geometrical parameters on the dynamic snap-through buckling and vibration characteristics of the post-buckled FG nanobeams.

Nonlinear static and dynamic isogeometric analysis of functionally graded microplates with graphene-based nanofillers reinforcement

The main contribution of this study is to propose an efficient and straightforward numerical model for exploring size-dependent geometrically nonlinear static and dynamic characteristics of functionally graded microplates reinforced by graphene nanofillers. To accomplish this purpose, modified couple stress theory including only one material length scale parameter is utilized to capture the size-dependent effect while the displacement field of small-scale structures is calculated by using four-variable refined plate theory and von-Kármán assumptions within the framework of isogeometric analysis. In this study, four dispersion patterns of graphene-based nanofillers along the thickness direction of microplate models are investigated under various static or dynamic loads. Solutions of nonlinear governing equation of motion can be obtained based on the Newton–Raphson iterative procedure and Newmark's time integration scheme. The Rayleigh damping is included, for the first time, to explore the influence of damping on the oscillations of functionally graded microplates reinforced by graphene nanofillers under three dynamic loads including step, triangular and half-sine pulses. Herein, several numerical examples are conducted to investigate the influences of a wide range of significant parameters on the geometrically nonlinear behaviors of functionally graded microplates with graphene nanofillers reinforcement. The significant outcomes of the present study can be referred to as benchmark results as well as provide valuable guides for designing small-scale structures with excellent features in the future.

Nonlinear dynamic analysis of sandwich shell panels with auxetic honeycomb core and curvilinear fibre reinforced facesheets

The nonlinear dynamic behaviour of sandwich shell panels having auxetic honeycomb core and facesheets composed of composite material with curvilinear fibres is investigated for the first time. A finite element study is performed considering a third-order shear deformation theory incorporating Murakami zig-zag effects for the analysis of the variable stiffness composite laminate (VSCL) auxetic honeycomb sandwich panels. A nine-noded C0 isoparametric quadratic shell element with eleven degrees of freedom per node is considered and von Kármán nonlinear strain-displacement equations are used to include the geometrical nonlinearity. The linear free vibration and nonlinear transient response of the panels are obtained using the eigenvalue solution and Newmark's time integration scheme, respectively. Most of the auxetic sandwich structures considered in the literature are made up of aluminium metal and therefore, the authors explored the viability of the VSCL based auxetic sandwich panels. A thorough numerical study including the effects of the geometrical parameters of the honeycomb unit cell, curvilinear fibre path angles and boundary conditions on the nonlinear dynamic response of auxetic sandwich panel with negative Poisson's ratio is presented.

A time domain element differential method for solving electromagnetic wave scattering and radiation problems

In this paper, a simple and efficient numerical method named time domain element differential method (TD-EDM) is proposed to solve electromagnetic wave scattering and radiation problems. In the method, the governing equations as well as boundary conditions are directly solved in a strong-form formulation. The computational domain is discretized using isoparametric elements, and the spatial derivatives of basic variables appearing in the governing equations are calculated by a set of analytical expressions of partial derivatives of the element shape functions with respect to global coordinates, and time derivatives of the basic variable are computed using the Newmark time integration scheme. The absorbing boundary condition (ABC) is applied on the truncation boundary to absorb the outgoing waves, which realizes the numerical modeling of an open infinite region. The incident wave is induced in the total-field region by setting equivalent electromagnetic currents on the total-field/scattering-field (TF/SF) boundary. The TD-EDM equations at the nodes located near the total-field boundary are modified to satisfy the conditions that the electric field values of all nodes involved belong to either the total-field or the scattered field. The near-to-far-field transformation technique is used to calculate the far scattered field, and then the target's Radar Cross Section (RCS) is evaluated. Several numerical results including the radiation from a line current source and the target's Radar Cross Section (RCS) are given to validate correctness of the proposed method.

Thermoelastic wave propagation due to local thermal shock on the functionally graded media

The present study is based on investigating the effect of locally applied thermal shock on the ceramic side of the functionally graded (FG) domain using nonlinear Lord-Shulman generalized thermoelastic theory. The constituent materials and therefore their properties in the two-dimensional FG domain are considered to be graded as a function of horizontal or vertical coordinates according to the power law. The finite element (FE) form of the nonlinear coupled differential equation is solved using the Picard method at each time-step in the framework of the Newmark time-integration scheme. The gradation of the material properties due to FGM is incorporated using graded finite elements in the FE formulation. The numerical results concerned with studying the effect of volume fraction index and orientation of the gradation of material properties across the FG domain are presented. It is observed from the study that both the gradation of material properties, as well as the direction of orientation of material gradation, have a substantial effect on the propagation of the thermal and elastic waves in the domain. Moreover, the thermal waves appear to be more dominant than the elastic waves at the time of application of thermal shock, which may also alter as time progresses.

An Enriched Finite Element Method with Appropriate Interpolation Cover Functions for Transient Wave Propagation Dynamic Problems

A novel enriched finite element method (EFEM) was employed to analyze the transient wave propagation problems. In the present method, the traditional finite element approximation was enriched by employing the appropriate interpolation covers. We mathematically and numerically showed that the present EFEM possessed the important monotonic convergence property with the decrease of the used time steps for transient wave propagation problems when the unconditional stable Newmark time integration scheme was used for time integration. This attractive property markedly distinguishes the present EFEM from the traditional FEM for transient wave propagation problems. Two typical numerical examples were given to demonstrate the capabilities of the present method.

Nonlocal nonlinear static/dynamic snap-through buckling and vibration of thermally post-buckled imperfect functionally graded circular nanoplates

The bi-stability characteristics of the post-buckled plates have demonstrated extensive potential applications for energy harvesting and vibration isolation. In this article, at first, thermal post-buckling, nonlinear bending, and vibration behavior of the functionally graded (FG) circular nanoplates in bifurcation buckling are investigated according to nonlocal elasticity theory. Then, the nonlinear static/dynamic snapping phenomena and free vibration responses of the thermally post-buckled nanoplate subjected to static and sudden types of mechanical load are presented. For this aim, the equations of motion in conjunction with the von-Kármán nonlinearity and geometrical imperfection are established in the framework of Hamilton's principle. In addition, two distinct cases of temperature distribution as well as two types of edge conditions are taken into consideration. The equations of motion are discretized using the Chebyshev-Ritz procedure along with three different numerical algorithms. To evaluate the nonlinear dynamic snap-through buckling, the Newmark time integration scheme is adopted. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. A set of parametric studies is presented to provide an insight into influences of the nonlocal parameter, geometrical imperfection, gradient index, and edge supports on the static and dynamic characteristics of the nanosystem.

Experimental verification of multi-fibre hybridization influence on dynamic deflection and stress values of curved composite panel

The finite-element time-dependent deflection and stress responses of the shallow composite panels subjected to variable mechanical loadings (uniformly distributed load and sinusoidally distributed load) are reported in this article. The study reveals the influence of the advanced fibre hybridization (glass–carbon–kevlar) on the dynamic responses and establishes the accuracy of the numerical responses by comparing them with the experimental values. The numerical steady-state deflections of the hybrid composite structure are evaluated using a generic mathematical model derived through Reddy's third-order shear deformation in conjunction with the finite-element technique and Newmark's time integration scheme. The experimentally evaluated mechanical properties of the fabricated composites are utilized in the computations. The validity of the computed solutions is ascertained with in-house experimental (transient deflection) results. Besides, the numerical model is extended to assess the parametric dependence (aspect ratio, thickness ratio, curvature ratio, geometry, hybrid schemes and support conditions) of the dynamic deflection/stress responses of hybrid composite shallow shell panels subjected to variable mechanical loading types.

Hybrid equilibrium element with high‐order stress fields for accurate elastic dynamic analysis

AbstractIn the present article the two‐dimensional hybrid equilibrium element formulation is initially developed, with quadratic, cubic, and quartic stress fields, for static analysis of compressible and quasi‐incompressible elastic solids in the variational framework of the minimum complementary energy principle. Thereafter, the high‐order hybrid equilibrium formulation is developed for dynamic analysis of elastic solids in the variational framework of the Toupin principle, which is the complementary form of the Hamilton principle. The Newmark time integration scheme is introduced for discretization of the stress fields in the time domain and dynamic analysis of both the compressible solid and quasi‐incompressible ones. The hybrid equilibrium element formulation provides very accurate solutions with a high‐order stress field and the results of the static and dynamic analyses are compared with the solution of the classic displacement‐based quadratic formulation, showing the convergence of the two formulations to the exact solution and the very satisfying performance of the proposed formulation, especially for analysis of quasi‐incompressible elastic solids.

Numerical analysis of nonlinear functionally graded piezoelectric energy harvester subjected to multi moving oscillators

In this paper, energy harvesting from nonlinear vibration of a functionally graded beam covered by a piezoelectric patch under multi-moving oscillators is studied. The material of both the substructure and the piezoelectric patch is assumed to be functionally graded in the thickness direction. A coupled system of equations considering Euler-Bernoulli beam theory and von-Karman nonlinearity as well as electromechanical coupling are derived using the generalized Hamilton’s principle. Finite element method as well as Newmark time integration scheme are used to solve the coupled nonlinear time dependent problem. The effects of different parameters including material distribution, velocity of the moving oscillators, piezoelectric patch thickness and load resistance on the output voltage and harvested power are investigated. Moreover, the effects of oscillator characteristics such as damping ratio and stiffness on the nonlinear behavior of the beam and harvested power are also studied. Results indicate that the aforementioned parameters have considerable effects on the harvested power. It is also shown that ignoring nonlinear effects may lead to erroneous and unacceptable results. To the best of authors’ knowledge, there is no study about energy harvesting from nonlinear vibration of beams under moving oscillators.

Piezoelectric energy harvesting from nonlinear dynamics of functionally graded plate under multi-moving loads and masses

In this study, energy harvesting from nonlinear vibrations of a functionally graded plate with bonded piezoelectric patch under moving loads and moving masses is investigated. The coupled nonlinear system of equations are derived using the generalized Hamilton’s principle considering both the geometrical nonlinearity and the nonlinear electromechanical coupling. Then, the equations are solved using finite element method and Newmark time integration scheme. For a case study, the effects of different parameters such as piezoelectric patch thickness, location of piezoelectric patch, different boundary conditions, and different width of plate on harvested power and the plate’s nonlinear vibration are investigated. This article is motivated by the lack of published studies on nonlinear distributed piezoelectric energy harvesters under moving loads (masses).

Seismic analysis of soil-structure interaction: Experimentation and modeling

This paper presents a simplified modelling strategy to simulate the soil-foundation-structure interaction under seismic loadings. The interaction of soil and structure is modeled by a macro-element with the coupling of geometric and material non-linearities. The model consists of 4 degrees of freedom in which the superstructure is lumped as a single degree of freedom (DOF) while the soil-foundation is modeled by 3 DOFs. The dynamic equilibrium equations are solved by a Newmark time integration scheme and implemented in Matlab. To verify the numerical model, an experimental investigation based on shaking table method has been conducted in the present study. Five series of earthquake motions with maximum acceleration increased from 0.1 m/s^2 to 1.4 m/s^2were applied and the results of time-dependent accelerations and displacements are extracted. Based on the result comparisons, it is found that the numerical results were well validated against the experimental results.

Trigonometric zigzag theory for free vibration and transient responses of cross-ply laminated composite plates

In this article, an application of the non-polynomial zigzag theory is presented to analytically derive the free vibration and transient responses of multilayered laminated composites and sandwich plates. The non-linear variations of the transverse shear stresses are incorporated in the mathematical model with the help of a trigonometric function, namely the secant function. The inter-laminar continuity conditions are artificially enforced at the layer interfaces that reduce the total number of field variables to five like the first-order shear deformation theory (FSDT). The present kinematic field yields more accurate results than the FSDT and also removes the requirement of any shear correction factor. The governing equilibrium equations of motion are obtained using Hamilton's principle and solved with Navier's solution technique. The responses from the coupled ordinary differential equations in time are obtained with the Newmark's time integration scheme. Numerical problems on the diaphragm supported laminated composites, and soft-core-sandwich plates are solved analytically to derive the free vibration and transient responses by taking into consideration the several geometrical and material features of the plate structures. The responses are also compared with the benchmark elasticity solutions and solutions reported in the literature from other shear deformation models. An exhaustive study on the transient analysis is presented by solving several problems of laminated composite plates that are subjected to various time-dependent loads and various forms of blast loads.

Transient analysis of thermo-mechanically shock loaded four-parameter power law functionally graded shells

The present work identifies the mechanical and thermal response of ceramic–metal composed functionally graded cylindrical and spherical shells of revolution exposed to rapidly applied mechanical and thermal loads on their inner surface. The gradation of the constituent materials is done in the thickness direction of the FG shells according to four-parameter power laws. The mechanical and thermal properties of the constituent materials are considered to be temperature-dependent. Mori-Tanaka model is employed to obtain effective material properties based on the homogenization method. A nonlinear 1-D heat conduction equation is solved in each time-step of unconditionally stable Crank-Nicolson time integration scheme. Thereafter, an uncoupled thermoelastic finite element code based on higher-order shear deformation theory for shells and Newmark time integration scheme is developed in MATLAB environment. Numerical results considering the effect of the geometry of the shell, thermal boundary conditions, temperature-dependent material properties, different parameters of FGM laws, and thermomechanical load on the response characteristic of shells are presented. It was found that the oscillations of a thermomechanical shocked shell initially deviated from its quasi-static nature of response but started to oscillate about the quasi-static curve as its mean position on removal of the mechanical load.

Superconvergent Isogeometric Transient Analysis of Wave Equations

A superconvergent isogeometric formulation is presented for the transient analysis of wave equations with particular reference to quadratic splines. This formulation is developed in the context of Newmark time integration schemes and superconvergent quadrature rules for isogeometric mass and stiffness matrices. A detailed analysis is carried out for the full-discrete isogeometric formulation of wave equations and an error measure for the full-discrete algorithm is established. It is shown that a desirable superconvergence regarding the isogeometric transient analysis of wave equations can be achieved by two ingredients, namely, the design of a superconvergent quadrature rule and the criteria to properly define the step size for temporal integration. It turns out that the semi-discrete and full-discrete isogeometric formulations of wave equations with quadratic splines share an identical quadrature rule for a sixth-order accurate superconvergent analysis. Meanwhile, the relationships between the time step size and the element size are presented for various typical Newmark time integration schemes, in order to ensure the sixth-order accuracy in transient analysis. Numerical results of the transient analysis of wave equations consistently reveal that the proposed superconvergent isogeometric formulation is sixth-order accurate with respect to spatial discretizations, in contrast to the fourth-order accuracy produced by the standard isogeometric approach with quadratic splines.

A coupled FEM–DEM method for the modeling of fluids laden with particles

This work presents a computational method for the solution of problems involving flowing fluid media laden with solid particles. The idea is based on previous works by the authors on (though then separately considered) fluid–structure interaction and particle dynamics. The fluid problem is treated through an Eulerian finite element approach, with the resulting system of nonlinear equations being iteratively solved by a Newton–Raphson procedure within a Newmark time integration scheme. The particle problem, in turn, is treated through a Lagrangian discrete element approach, wherein both particle-to-particle and particle-to-wall (rigid surfaces) contacts are fully permitted and resolved. The influence of the fluid on the motion of the particles is represented by means of forces and moments, which are computed from solution of the fluid flow around the particles and imposed on the latter in a coupled iterative way. The influence of the particles on the fluid, in turn, is considered by imposing consistent boundary conditions on the fluid at its corresponding interfaces with the particles. This is achieved through an immersed boundaries technique. An implicit, staggered, coupled FEM–DEM scheme is developed within a time-marching solution process. Examples of numerical simulations are provided to illustrate the applicability and potentialities of the proposed method.