Newmark Time Integration Research Articles

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.

A Macro-element for Modeling the Non-linear Interaction of Soil-shallow Foundation under Seismic Loading

This paper presents a macro-element for simulating the seismic behavior of the soil- shallow foundation interaction. The overall behavior in the soil and at the interface is replaced by a macro-element located at the base of the superstructure. The element reproduces the irreversible elastoplastic soil behavior (material non-linearity) and the foundation uplift (geometric non-linearity) at the soil- foundation interface. This new macro-element model with three degrees-of-freedom describes the force-displacement behavior of the footing center. The single element is restrained by the system of equivalent springs and dashpots. The footing is considered as a rigid body. It is solved by a suitable Newmark time integration scheme and implemented in Matlab to simulate the nonlinear behavior of soil-shallow foundation interaction under seismic loading. A reduce scaled soil-foundation system has been tested on a shaking table at the University of Transport and Communications, Hanoi, Vietnam. Five series of earthquake motions were used with maximum acceleration increased from 0.5 to 2.5 . The comparison of numerical results obtained from the simulation and experimentations shows the satisfactory agreement of the model. The proposed macro-element can be used to predict the seismic behavior of a wider variety of configurations.

An efficient numerical method to analyze low‐velocity impact response of carbon fiber reinforced thermoplastic laminates

AbstractLow velocity impact behavior of carbon fiber reinforced thermoplastic laminates are analyzed in this article. Governing equations are derived based on the first‐order shear deformation theory and Hamilton's principle. The impact contact force history between the impactor and laminate is formulated by the Hertzian contact law. A Ritz‐based solution in space domain for various boundary conditions is developed. Solutions of the system equations in time domain are solved by the Newmark's time integration scheme. Impact parameters such as impact force history, indentation history, and contact time are considered. The accuracy of this method is validated by comparing the obtained results with those from the ABAQUS/Explicit software simulation. The effects of impact velocity, plate geometrical parameters, stacking sequence, and temperature on the dynamic response of laminates under clamped boundary condition and support boundary condition were discussed.

Transient responses of two mutually interacting single-walled boron nitride nanotubes induced by a moving nanoparticle

This paper reports a theoretical investigation on the dynamic behaviour of two mutually interacting single-walled boron nitride nanotubes (BNNTs) under a moving nanoparticle. Both the electromechanical coupling effect and nonlocal effect are considered by employing respectively the theory of piezoelectricity and nonlocal elasticity. The interaction between the two BNNTs is considered by calculating the van der Waals (vdW) force between atoms on the respective nanotubes. The Hamilton's principle is then used to derive the governing equations of motion based on the Rayleigh and Timoshenko beam model. The incremental harmonic balance method, Galerkin method and the Newmark time integration method are jointly employed to numerically obtain the transient vibration responses of the nanotubes. The influences of the size effect, aspect ratio, inertial force, electrical potential, as well as the mass and velocity of the nanoparticle on the transient responses are discussed.

Large Deflection of Elastic Beams under Impact by Rigid Particles

The structural response of a slender beam subjected to impact by several rigid particles is investigated. A higher-order shear deformation beam model and the von-Karman geometric measure are employed. The equations of motion, continuity conditions at the points of impact by particles, and geometric and natural boundary conditions for the beam are derived by means of Hamilton’s principle. The generalized differential quadrature and Newmark time integration methods are utilized to carry out the numerical solution of the system of nonlinear partial differential equations comprised of the equations of motion and stress resultants. The durations of contact, displacements, and stress resultants are obtained for beams with various types of boundary conditions.

Active control of functionally graded carbon nanotube–reinforced composite plates with piezoelectric layers subjected to impact loading

To the best of the authors’ knowledge, this is the first attempt in the open literature to study the active control of the dynamic response of functionally graded carbon nanotube–reinforced composite plates with piezoelectric layers, as target composite plates, subjected to impact loading. The theoretical formulation of the composite plates with piezoelectric layers is developed using the element-free improved moving least-squares Ritz model and the higher-order shear deformation theory. The effective material properties of the carbon nanotube–reinforced composite layer are estimated by the Mori–Tanaka method. The modified nonlinear Hertz contact law is used to identify the contact force between the target composite plates and the spherical impactor. The Newmark time integration method is utilized to calculate the resulting dynamic response. A constant velocity feedback controller is efficiently used for the active control of the dynamic response of the target plates subjected to impact loading. The results revealed that the current model can successfully reduce and suppress the resulting displacement caused by impact loading. Additionally, the effects of some passive configurations on the target plates’ dynamic response are presented. The effect of altering both active and passive control configurations together on the target plates’ dynamic response is discussed as well.

A complete Galerkin’s type approach of finite element for the solution of a problem on modified Green–Lindsay thermoelasticity for a functionally graded hollow disk

The present work is concerned with the modified Green–Lindsay thermoelasticity theory involving strain-rate. This theory has been proposed very recently to modify the Green–Lindsay (GL) model of thermoelasticity by introducing both temperature and strain-rate in the constitutive relations of coupled thermo-mechanics. We consider a problem involving coupled thermo-mechanical interactions inside a functionally graded hollow disk due to thermal shock applied at its stress-free inner and outer boundaries. The material properties of the disk are assumed to be non-homogeneous and changing along the radial direction according to a volume fraction rule with a power of non-homogeneity index term. We formulate the problem by considering the basic governing equations of GL and modified GL thermoelasticity theories in a unified form and derive a linear system of coupled partial differential equations with variable coefficients. The major aim of this work is to apply a complete finite element method to obtain the solution of the problem. We solve the system of equations by applying a Galerkin’s approach of FEM for the space domain and derive the time differential system of equations. To solve this time differential system of equations, we use two different approaches: (1) a Galerkin’s type FE approach and (2) the Newmark time integration scheme. We compare the results obtained from these two different approaches and find that the solution obtained by complete Galerkin’s approach of FEM perfectly matches with the corresponding solution obtained from Newmark time integration scheme. We further compare the CPU time taken by these two methods with the CPU time taken by the trans-finite element method and reveal the efficiency of the present method over trans-FE method. The variation of different physical field variables with space and time is discussed for various values of the non-homogeneity index by highlighting the difference in the results under the GL model and modified GL model.

Isogeometric nonlinear transient analysis of porous FGM plates subjected to hygro-thermo-mechanical loads

Abstract To provide reference solutions and results for structural and material design, nonlinear transient responses of porous functionally graded plate (PFGM) in hygro-thermo-mechanical environments are studied. Two different porous distributions through the thickness are considered. The material properties such as Young's modulus, Poisson's ratio and thermal conductivity are computed by a modified power law. The hygro-thermal effects are considered as nonlinear through the thickness of the plate. The geometrically nonlinear transient behaviors are expressed by adopting the von Karman relations and solved by Newmark time integration scheme. Based on a combination between the third-order shear deformation theory (TSDT) and isogeometric analysis (IGA), discretize governing equations are approximated. These approaches achieve naturally any desired degree of continuity of basis functions, so that they are easy to fulfil the C1-continuity requirement of the plate model. The formulations take into account the transverse shear deformation and account for the material properties at elevated moisture concentrations and temperature. The effects played by the moisture concentration, temperature rise, porous volume fraction, boundary conditions and thickness-to-length ratio are performed and results illustrate interesting dynamic phenomenon for PFGM in hygro-thermo-mechanical environments. With obtained results, the nonlinear characteristics of the proposed structure with porosities are based on physical parameters.

Stochastic low-velocity impact analysis of sandwich plates including the effects of obliqueness and twist

Abstract This paper quantifies the influence of uncertainty in the low-velocity impact responses of sandwich plates with composite face sheets considering the effects of obliqueness in impact angle and twist in the plate geometry. The stochastic impact analysis is conducted by using finite element (FE) modelling based on an eight nodded isoparametric quadratic plate bending element coupled with multivariate adaptive regression spline (MARS) in order to achieve computational efficiency. The modified Hertzian contact law is employed to model contact force and other impact parameters. Newmark's time integration scheme is used to solve the time-dependent equations. Comprehensive deterministic as well as probabilistic results are presented by considering the effects of location of impact, ply orientation angle, impactor velocity, impact angle, face-sheet material property, twist angle, plate thickness and mass of impactor. The relative importance of various input parameters is determined by conducting a sensitivity analysis. The results presented in this paper reveal that the impact responses of sandwich plates are significantly affected by the effect of source-uncertainty that in turn establishes the importance of adopting an inclusive stochastic design approach for impact modelling in sandwich plates.

Wavelet based damage identification and dynamic pull-in instability analysis of electrostatically actuated coupled domain microsystems using generalized differential quadrature method

Abstract In this study, the effects of edged-crack on dynamic pull-in instability of fixed-fixed microbeams under suddenly applied electrostatic excitations and nonlinear squeeze film damping are investigated. The deformable electrode is modeled based on size dependent Euler–bernoulli beam formulations with modified couple stress theory. Also fluid interaction with electrode is modeled by nonlinear Reynolds equation. The model accounts for the von Karman nonlinear strains and edge crack in the domain. The edged crack is modeled as a massless torsional spring and is incorporated in the continuity equations of micro-beam formulations. Employing the Hamilton's principle, the governing nonlinear equations are attained and are then discretized by generalized differential quadrature method. This approach is combined with Newmark time integration and direct iteration method and is employed to investigate dynamic pull-in instability of the microsystem. The stability analysis of the cracked microbeam is also implemented by evaluating the largest Lyapunov exponent, the sign of which describes the character of the time dependent response. Wavelet transform based on Gabor function is applied for identifying the damage in microstructure. The location of damage is determined from the sudden peaks in the spatial variation of the transformed response of damped microbeam by this algorithm. Finally, a comprehensive study is carried out to examine the influence of the crack depth and crack position on pull-in instability characteristics of microbeams. Also, the efficiency of proposed wavelet analysis for damage detection in microstructures is assessed.

Isogeometric collocation for implicit dynamics of three-dimensional beams undergoing finite motions

We propose a novel approach to the implicit dynamics of shear-deformable geometrically exact beams, based on the isogeometric collocation method combined with the Newmark time integration scheme extended to the rotation group SO(3). The proposed formulation is fully consistent with the underlying geometric structure of the configuration manifold. The method is highly efficient, stable, and does not suffer from any singularity problem due to the (material) incremental rotation vector employed to describe the evolution of finite rotations. Consistent linearization of the governing equations, variables initialization and update procedures are the most critical issues which are discussed in detail in the paper. Numerical applications involving very large motions and different boundary conditions demonstrate the capabilities of the method and reveal the critical role that the high-order approximation in space may have in improving the accuracy of the solution.

A scaled boundary finite element formulation for dynamic elastoplastic analysis

SummaryThis study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.

Nonlinear periodic response of bimodular laminated composite annular sector plates

The steady state non-linear response of bimodular composite laminated annular sector plates is investigated using the field consistent eight-noded finite element based on the first-order shear deformation theory and Bert's constitutive model. The periodic forced response is obtained using the modified shooting method and arc-length/pseudo arc-length continuation techniques. Within a shooting cycle, the solution of the governing equation is obtained using Newmark's time integration coupled with Newton Raphson method. The effects of bimodularity, geometric nonlinearity, boundary conditions, load amplitude, lamination scheme and sector angle on the response characteristics are presented. Significantly large difference in the peak amplitudes is predicted with and without geometric nonlinearity. The higher harmonic contributions in the steady state displacement/stresses are demonstrated using frequency spectra and phase plane plots. Through the strain energy plots, the participation of even and odd order harmonics is correlated to the quadratic and cubic restoring forces in a cycle. The total number of degrees of freedom for converged results is 805 for CCCC and 865 for CSCS/SCSC plates resulting in a system seldom treated in the literature on nonlinear steady state periodic response and the results presented may serve as reference for validation of approximate solutions.

An isogeometric Bézier finite element analysis for piezoelectric FG porous plates reinforced by graphene platelets

In this study, we for the first time present an isogeometric Bézier finite element formulation for bending and transient analysis of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs) embedded in piezoelectric layers. We name it as PFGP-GPLs for short. The plates are constituted by a core layer, which contains the internal pores and GPLs dispersed in the metal matrix either uniformly or non-uniformly according to three different patterns, and two piezoelectric layers perfectly bonded on the top and bottom surfaces of host plate. The modified Halpin–Tsai micromechanical model is used to estimate the effective mechanical properties which vary continuously along thickness direction of the core layer. In addition, the electric potential is assumed to vary linearly through the thickness for each piezoelectric sublayer. A generalized C0-type higher-order shear deformation theory (C0-HSDT) in association with isogeometric analysis (IGA) based on Bézier extraction is investigated. Our approach allows performing all computations the same as in the conventional finite element method (FEM) yet the present formulation shows more advantages. The system of time-dependent equations is solved by the Newmark time integration scheme. The effects of weight fractions and dispersion patterns of GPLs, the coefficient and distribution types of porosity as well as external electrical voltages on structure’s behaviors are investigated through several numerical examples. These results, which have not been published before, can be considered as reference solutions for future works.

Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis

The study using numerical methods on porous functionally graded (FG) nanoplates is still somewhat limited. This paper focuses on porosity-dependent nonlinear transient analysis of FG nanoplates using isogeometric finite element approach. In order to capture the small size effects, the Eringen's nonlocal elasticity based on higher order shear deformation theory (HSDT) are used to model the porous FG nanoplates. Two distributions of porosities inside FG materials are incorporated and defined via a modified rule of mixture. The nonlinear transient nonlocal governing equations under transverse dynamic loads are formulated by using the von Kármán strains and are solved by Newmark time integration scheme to obtain geometrically nonlinear responses. It is indicated that nonlinear transient deflections of the porous FG nanoplate are significantly influenced by material composition, porosity, nonlocal parameters, volume fraction exponent, porosity distributions, geometrical parameters and dynamic load characteristics.

Vibrational control scrutiny of physically affected SWCNT acted upon by a moving nanoparticle in the framework of nonlocal–strain gradient theory

The possibility of moving drug materials into the intelligent nano-materials such as nanotubes such as set of DNA or RNA molecules to change the behavior of cells is an important problem in the nano-medicine science. This paper deals with vibrational control of magnetically thermally affected single-walled carbon nanotube (SWCNT) under a moving nanoparticle using the nonlocal–strain gradient theory based on the Rayleigh beam model. The elastic medium is modeled as Pasternak substrate. A gain matrix with time-varying behaviors and displacement–velocity feedback in the framework of linear classical optimal control procedure is used to suppress the vibration responses of the SWCNT. Hamilton, Galerkin and Newmark time integration principles are jointly utilized to ascertain the equations of motion. The influences of the nonlocal and material length-scale parameters, the velocity of nanoparticle and physical fields on the vibration behavior of the SWCNT are explored. Likewise, a specified control algorithm in suppressing the vibrational behavior of SWCNT under the effect of moving nanoparticle is examined.

Stochastic low-velocity impact on functionally graded plates: Probabilistic and non-probabilistic uncertainty quantification

This paper quantifies the compound effect of source-uncertainties on low-velocity impact of functionally graded material (FGM) plates following a coupled surrogate based finite element simulation approach. The power law is employed to evaluate the material properties of FGM plate at different points, while the modified Hertzian contact law is implemented to determine the contact force and other parameters in a stochastic framework. The time dependent equations are solved by Newmark's time integration scheme. Insightful results are presented by investigating the effects of degree of stochasticity, oblique impact angle, thickness of plate, temperature, power law index, and initial velocity of impactor following both probabilistic and non-probabilistic approaches along with in-depth deterministic analyses. A detail probabilistic analysis leading to complete probabilistic characterization of the structural responses can be carried out when the statistical distribution of the stochastic input parameters are available. However, in many cases concerning FGM, these statistical distributions may remain unavailable due to the restriction of performing large number of experiments. In such situations, a fuzzy-based non-probabilistic approach could be appropriate to characterize the effect of uncertainty. A surrogate based approach based on artificial neural network coupled with the finite element model for low-velocity impact analysis of FGM plates is developed for achieving computational efficiency. The numerical results reveal that the low-velocity impact on FGM plates is significantly influenced by the effect of inevitable source-uncertainty associated with the stochastic system parameters, whereby the importance of adopting an inclusive design paradigm considering the effect of source-uncertainties in the impact analysis is established.

Frequency domain modeling of nonlinear end stop behavior in Tuned Mass Damper systems under single- and multi-harmonic excitations

Nonsmooth dynamics of a Tuned Mass Damper system with lateral stops are studied using an alternating frequency/time harmonic balancing (AFT-HB) method. To this end, an extremely stiff end stop nonlinearity is considered. The application range of AFT-HB is investigated by including up to 250 harmonics in the external force, as well as in the motion description. Numerical simulations are performed by making use of a Newmark time integration algorithm for numerical verification of the results. The results for single harmonic excitations are further verified with an existing pseudo-arclength path-following tool. Two excitation scenarios are considered: single harmonic- and a wide-spectrum excitation with uniform distribution and random phase correlation between the harmonics. The AFT-HB algorithm is found to accurately reproduce the time integration results, for all considered cases. Finally, insights are gained into the differences between the system responses to single- and multi-harmonic excitations.

Structural damage detection using time domain responses and an optimization method

ABSTRACTAn efficient method is proposed to determine the location and severity of structural damage using time domain responses and an optimization method. The time domain responses utilized here are the nodal accelerations measured at the limited points of a structure subjected to an impulse load. The nodal accelerations of the structure are obtained by Newmark time integration method. Firstly, using nodal accelerations extracted for the damaged structure and an analytical model of the structure, an objective function is defined for optimization. Then, the optimization-based damaged detection problem is solved via a differential evolution algorithm for finding the location and severity of damage. In order to assess the accuracy of the proposed method, four numerical examples are considered. Simulation results reveal the efficiency of the method for properly identifying damage with considering measurement noise.

A Ritz Procedure for Transient Analysis of Dam–Reservoir Interaction

A simple and accurate variational formulation is proposed to study the transient dynamic responses of two-dimensional dam–reservoir system. The proposed methodology uses a simple weak formulation with Ritz procedure to reduce the governing partial differential equations of the fluid and structure to a set of ordinary differential equations. An analog procedure is then presented to exactly implement the natural boundary condition of the problem at fluid–structure interface. The Newmark time integration scheme is used to solve the resulting system of ordinary differential equations. To demonstrate the accuracy and efficiency of the proposed formulation, the problem is also solved using the finite element method (FEM). It is found that the proposed method can produce better accuracy than the FEM using less computational time. The technique presented in this investigation is general and can be used to solve various fluid–structure interaction problems.