Direct Time Integration Method Research Articles

Dynamic topology optimization of piezoelectric structures with active control for reducing transient response

This paper investigates topology optimization of the piezoelectric actuator/sensor coverage attached to a thin-shell structure to improve the active control performance for reducing the dynamic response under transient excitations. The constant gain velocity feedback (CGVF) control algorithm is considered and the structural dynamic response under the corresponding active damping effect is evaluated with a direct time integration method. In the mathematical formulation of the considered topology optimization model, the time integral of the displacement response over a specified time interval of interest is taken as the objective function. The pseudo-densities describing the piezoelectric material distribution are taken as the design variables, and a penalization model on the structural stiffness and piezoelectric effect is employed. The adjoint-variable sensitivity analysis scheme for a general integral function within a given time interval is derived, which facilitates a gradient-based mathematical programming solution of the optimization problem. Numerical examples demonstrate that the proposed method can generate meaningful optimal topologies of piezoelectric layers. Also, the influences of the control gain and the integration time intervals in the objective function on the optimal solutions are discussed. The proposed method can be used for providing useful guidance to the layout design of the actuator/sensor layers attached to a thin-shell structure subject to dynamic excitations, in particular impact forces. It is also confirmed that the achieved vibration reduction is mainly due to improvement of the active control performance rather than changes of the structural dynamic stiffness/mass property.

Geometrically nonlinear dynamic analysis of thin shells by a four-node quadrilateral element with in-plane rotational degree of freedom

A simple and effective finite element incremental formulation based on the updated lagrangian corotational description for geometrically nonlinear dynamic analysis of shell structure is presented in this work. The flat shell element used is the classical four-node quadrilateral DKQ shell finite element, combined with the improvements of the in-plane behaviour by incorporation of the drilling rotation degree of freedom. The main goal is to have a good flat shell element of quadrilateral geometry, leading to reliable solutions in linear and geometrically nonlinear dynamic analysis. The Newmark’s direct time integration method is adopted to integrate the equations of motion, while the Newton–Raphson method is used for iterating within each time step increment until equilibrium is achieved. The results obtained through a series of selected examples demonstrate the effectiveness of the used shell finite element to predict the nonlinear dynamic response of complex structures, and the robustness of this nonlinea...

Computational Method of the Dynamic Response for Nonviscously Damped Structure Systems

AbstractNonviscous damping models in which the damping forces depend on the history of velocities via convolution integrals over some kernel functions have risen in many different subjects. The aim of this paper is to extend the Newmark method from viscously damped structure systems to nonviscously damped structure systems. The proposed analysis method for nonviscously damped structure systems is derived based on Newmark’s assumption of acceleration. The convolution integral term is calculated directly using the trapezoidal rule. The computational procedure of the proposed direct time integration method for nonviscously damped structure systems is given in detail. Finally, the dynamic responses of two nonviscously damped structure systems are computed using the proposed method. The accuracy and efficiency of the proposed method are discussed through comparison with another developed method.

Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution

This article presents a space-time adaptive strategy for transient elastodynamics. The method aims at computing an optimal space-time discretization such that the computed solution has an error in the quantity of interest below a user-defined tolerance. The methodology is based on a goal-oriented error estimate that requires accounting for an auxiliary adjoint problem. The major novelty of this paper is using modal analysis to obtain a proper approximation of the adjoint solution. The idea of using a modal-based description was introduced in a previous work for error estimation purposes. Here this approach is used for the first time in the context of adaptivity. With respect to the standard direct time-integration methods, the modal solution of the adjoint problem is highly competitive in terms of computational effort and memory requirements. The performance of the proposed strategy is tested in two numerical examples. The two examples are selected to be representative of different wave propagation phenomena, one being a 2D bulky continuum and the second a 2D domain representing a structural frame.

Seismic response analysis of an oil storage tank using Lagrangian fluid elements

Three-dimensional Lagrangian fluid finite element is applied to seismic response analysis of an oil storage tank with a floating roof. The fluid element utilized in the present analysis is formulated based on the displacement finite element method considering only volumetric elasticity and its element stiffness matrix is derived by using one-point integration method in order to avoid volumetric locking. The method usually adds a rotational penalty stiffness to satisfy the irrotational condition for fluid motion and modifies element mass matrices through the projected mass method to suppress spurious hourglass-mode appeared in compensation for one-point integration. In the fluid element utilized in the present paper, a small hourglass stiffness is employed. The fluid and structure domains for the objective oil storage tank are modeled by eight-node solid elements and four-node shell elements, respectively, and the transient response of the floating roof structure or the free surface are evaluated by implicit direct time integration method. The results of seismic response analyses are compared with those by other method and the validation of the present analysis using three-dimensional Lagrangian fluid finite elements is shown.

Static and Dynamic Characteristics of Composite Conoidal Shell Roofs

A thorough review of the existing literature reflects that forced vibration studies of laminated composite conoidal shells with complicated boundary conditions are missing. Hence, the present paper aims to fill the lacuna. A finite element code utilizing eight-noded doubly curved elements together with modified Sanders’ first approximation theory for thin shells is used to study the forced vibration behavior of moderately thin laminated composite conoidal shells subjected to three different uniformly distributed time-dependent forces. Newmark’s direct time integration method is used to solve the dynamic problem. Results obtained using the present code are compared with the values available in the literature, and a good agreement of the results confirms the accuracy of the proposed code. The transient responses of the laminated shell are studied meticulously for parametric variations like boundary conditions and stacking orders of cross and angle-ply laminates and are compared with bending responses of the shell to conclude on the necessity of the dynamic study.

A new direct time integration method for the equations of motion in structural dynamics

AbstractA direct time integration method is presented for the solution of the equations of motion describing the dynamic response of structural linear and nonlinear multi‐degree‐of‐freedom systems. It applies also to large systems of second order differential equations with fully populated, non symmetric coefficient matrices as well as to equations with variable coefficients. The proposed method is based on the concept of the analog equation, which converts the coupled N equations into a set of single term uncoupled second order ordinary quasi‐static differential equations under appropriate fictitious loads, unknown in the first instance. The fictitious loads are established from the integral representation of the solution of the substitute single term equations. The method is simple to implement. It is self starting, unconditionally stable and accurate and conserves energy. It performs well when large deformations and long time durations are considered and it can be used as a practical method for integration of the equations of motion in cases where widely used time integration procedures, e.g. Newmark's, become unstable. Several examples are presented, which demonstrate the efficiency of the method. The method can be straightforward extended to evolution equations of order higher than two.

Transient response of a bi-layered multiferroic composite plate

In this paper, a three-dimensional finite-element formulation for the multiferroic composite is developed and implemented into the commercial software ABAQUS for its transient analysis. First, a special three-dimensional eight-node solid element is designed to handle the multiferroic composite made of elastic, piezoelectric, and piezomagnetic materials. Second, a user-defined subroutine for this newly developed element is implemented into ABAQUS. Finally, the transient responses of a bi-layered multiferroic composite are calculated by using the direct time integration method. Two typical magnetic potential signals, Gauss and Ricker pulses, are applied to the composite with various time durations of excitation. The induced electric field shows that the transient response can be substantially influenced by the input signal, which could be tuned for the strongest electric output.

Dynamic characteristics of fluid-conveying functionally graded cylindrical shells under mechanical and thermal loads

Abstract Considering rotary, in-plane inertias, and fluid velocity potential, the dynamic characteristics of fluid-conveying functionally graded materials (FGMs) cylindrical shells subjected to dynamic mechanical and thermal loads are investigated, where material properties of FGM shells are considered as graded distribution across the shell thickness according to a power-law, and dynamic thermal loads applied on the shell is considered as non-linear distribution across the thickness of the shell. The linear response characteristics of fluid-conveying FGM cylindrical shells are obtained by using modal superposition and Newmark’s direct time integration method.

Multiharmonic Response Analysis of Systems With Local Nonlinearities Based on Describing Functions and Linear Receptance

Direct time integration methods are usually applied to determine the dynamic response of systems with local nonlinearities. Nevertheless, these methods are computationally expensive to predict the steady state response. To significantly reduce the computational effort, a new approach is proposed for the multiharmonic response analysis of dynamical systems with local nonlinearities. The approach is based on the describing function (DF) method and linear receptance data. With the DF method, the kinetic equations are converted into a set of complex algebraic equations. By using the linear receptance data, the dimension of the complex algebraic equations, which should be solved iteratively, are only related to nonlinear degrees of freedom (DOFs). A cantilever beam with a local nonlinear element is presented to show the procedure and performance of the proposed approach. The approach can greatly reduce the size and computational cost of the problem. Thus, it can be applicable to large-scale systems with local nonlinearities.

Modélisation par éléments finis du chauffage infrarouge des membranes thermoplastiques semi-transparentes

Resume Dans cette etude, nous avons etabli par la methode des elements finis en trois dimensions une approche simplifiee pour analyser l'evolution de la temperature au sein d'une membrane semi-transparente de type polyethylene terephtalate (PET) amorphe, exposees a une source radiative. En se basant sur le comportement optique du PET, un transfert radiatif monodirectionnel, suivant l'epaisseur de la membrane est considere. Des simulations numeriques ont permis de confronter et de valider les resultats obtenus vis-a-vis des resultats analytiques et experimentaux. Comme application, nous avons etudie le chauffage d'une feuille thermoplastique en PET exposee a une source radiative infrarouge.

Dynamic stability of a semi-circular pipe conveying harmonically oscillating fluid

The dynamic stability of a semi-circular pipe conveying harmonically oscillating fluid is investigated in this study. For analysis, harmonically oscillating flow is regarded as a parametrically excited system with time-varying coefficients. Considering the extensibility and nonlinearity of the semi-circular pipe, the equations of motion and the associated boundary conditions are obtained by using the extended Hamilton principle. Applying Floquet theory to the derived equations, the stability of the pipe is analyzed for variations of the amplitude and oscillating frequency of the fluid velocity. The effects of the mean value for the fluid velocity on the stability are also analyzed. It is found that the semi-circular pipe may become unstable when the fluctuation frequency approaches some multiples or sum of the out-of-plane natural frequencies. Furthermore, it is observed that the instability regions increase with the mean value of the velocity. The results obtained via the stability analysis are verified by time responses computed by using a direct time integration method.

Formulation of Equations of Motion for a Simply Supported Bridge under a Moving Railway Freight Vehicle

Based on energy approach, the equations of motion in matrix form for the railway freight vehicle-bridge interaction system are derived, in which the dynamic contact forces between vehicle and bridge are considered as internal forces. The freight vehicle is modelled as a multi-rigid-body system, which comprises one car body, two bogie frames and four wheelsets. The bogie frame is linked with the car body through spring-dashpot suspension systems, and the bogie frame is rigidly linked with wheelsets. The bridge deck, together with railway track resting on bridge, is modelled as a simply supported Bernoulli-Euler beam and its deflection is described by superimposing modes. The direct time integration method is applied to obtain the dynamic response of the vehicle-bridge interaction system at each time step. A computer program has been developed for analyzing this system. The correctness of the proposed procedure is confirmed by one numerical example. The effect of different beam mode numbers and various surface irregularities of beam on the dynamic responses of the vehicle-bridge interaction system are investigated.

HLI, A Direct Method Suitable for Partial and Fully Implicit Time Integration of Primitive Equation Meteorological Models

HLI, the homogeneously linearized implicit method is a direct implicit time integration method for nonlinear prognostic equations. For homogeneous linear test problems it is identical to the known implicit method, when applied to the full model area. As most theoretical results concerning the implicit method use linear homogeneous test problems, these apply immediately to HLI. The implicit method, when applied to the whole model area, is called the fully implicit method. In order to reduce the numerical expense for large integration areas, a partial implicit method is defined by posing artificial numerical boundary conditions, which result in a smaller set of equations, corresponding to the part of the integration area chosen. The application of either full or partial methods to nonlinear and inhomogeneous problems is achieved by defining for each grid point the local homogeneously linearized problem. Partial schemes are convenient when using multiprocessor computers, as only local and neighbouring data have to be used at each grid point. Only grid point discretisations of finite order are admitted. The proposed method would not be suitable for the spectral method. Computational examples are presented to show the suitability of this method for inhomogeneous situations and the ease of the treatment of internal boundary conditions. A semi-implicit treatment was also tested, which treats only the terms responsible for the fast waves implicit. This requires only the Fourier transformation of one field and in the generalised Fourier back transformation only one linear equation has to be solved, rather that a system of equations. In this paper only the trapezoidal implicit method was used. HLI could, however, be applied in the same way with other implicit methods.

Modal Coordinate Formulation for a Simply Supported Bridge Subjected to a Moving Train Modelled as Two-Stage Suspension Vehicles

Modal coordinate formulation for analysing the dynamic interaction between a moving train and a simply supported bridge is presented in this article. The train is composed of a series of identical vehicles, and each vehicle is modelled as a four-wheelset mass-spring-damper multi-rigid body system with two-stage suspension having ten degrees of freedom (DOFs). A simply supported bridge, together with the track, is modelled as a Bernoulli-Euler beam. The deflection of the beam is described by superimposing modes. The train and the beam are regarded as an entire dynamic system, and then the modal coordinate formulation with time-dependent coefficients for this system is directly derived from the principle of virtual work. The formulation is solved by direct time integration method, to obtain the dynamic responses of this system. The correctness of the proposed formulations is illustrated by a comparison with the existing literature. The formulation helps save computer time using a few beam modes for analysing the dynamic responses of an entire train-bridge interaction system. The proposed formulation can also be applied to analyse the dynamic responses of a simply supported bridge subjected to a moving train modelled as two-wheelset four DOFs vehicles. Two numerical examples are given for illustrating the applications of the proposed formulation.

A Fractional Derivative Viscoelastic Model for Hybrid Active-Passive Damping Treatments in Time Domain - Application to Sandwich Beams

This work presents a finite element formulation for the dynamic transient analysis of a damped adaptive sandwich beam composed of a viscoelastic core and elastic-piezoelectric laminated faces. The latter are modeled using the classical laminate theory, which takes the electromechanical coupling into account by modifying the stiffness of the piezoelectric layers. For the core, a fractional derivative model is used to characterize its viscoelastic behavior. Equations of motion are solved using a direct time integration method based on the Newmark scheme in conjunction with the Grunwald approximation of fractional derivatives. Emphasis is given to the finite element implementation of the fractional derivative model and to the influence of the electromechanical coupling.

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler–Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.

Efficient techniques for predicting viscoelastic behavior of sublaminates

Even for linear elastic behavior, stress analysis of thick laminated composites can be very computation intensive if every lamina is modeled discretely. In such cases, modeling of individual lamina is impractical and the homogenization method for sublaminates becomes essential. In the current work, 3D homogenization formulas for an elastic sublaminate, which were derived by the authors in previous work, were utilized to determine the 3D effective properties for a viscoelastic sublaminate. The properties were determined by three methods that exploited the 3D elastic homogenization formulas: (i) quasi-elastic method, (ii) correspondence principle, and (iii) direct time integration of the incremental viscoelastic equations. The finite element method with discrete modeling of the plies was used to obtain reference solutions. The effective viscoelastic properties obtained using the three methods based on the elastic homogenization formulas were in very good agreement with the reference solution. Among these methods, the quasi-elastic method was found to be both accurate and the simplest method in determining the effective properties. The methods were also used to predict the stress response of a sublaminate to different strain histories. The direct time integration method using the 3D elastic homogenization formulas performs accurately and efficiently for this problem.

Maglev Vehicle/Guideway Vertical Random Response and Ride Quality

Summary The research and development (R & D) of maglev technology had made a great progress in China since the early 1980s. Especially, a 35 km-long Shanghai high-speed maglev railway employing the German Transrapid system began to be constructed on March 1, 2001. Based on the Transrapid system, the paper develops a 10-degree-of-freedom model of maglev vehicle running over three types of guideways with constant speed. Random guideway irregularities are discussed and taken into account for simulation of the vehicle response and for evaluation of the ride comfort. Using the direct time integration method and the discrete fast Fourier transform (DFFT), random responses of the maglev vehicle-guideway systems are obtained and analyzed. Numerical results show that the resonant frequency of car body acceleration response is 0.5–1 Hz, and there is a 2.2 Hz periodic vibration due to the periodic configuration of rigid piers when the maglev vehicle travels over the elevated-beam guideway. The car body acceleration power spectral density (PSD) curves meet well the ride quality criterion of the urban tracked aircushion vehicle (UTACV) and the maximum acceleration of car body is less than 0.05 g. Moreover, the Sperling ride index values are less than 2.5 as long as the operational speed is less than 450 km/h. It is concluded that the maglev vehicle ride quality is quite well.

Finite Element Analysis of Low-Velocity Impact Damage in Composite Laminates

This paper investigates the low-velocity impact behaviour and impact-induced damages in graphite/epoxy composite laminates. A three-dimensional finite element and transient dynamic analysis is performed to calculate the time-varying displacements, forces, strains and stresses throughout the laminate resulting from transverse impact. A layered version of an eight-noded isoparametric brick element with incompatible modes is used to model the laminate. Transient dynamic equilibrium equation is integrated step-by-step with respect to time using Newmark direct time integration method. Modified Hertzian contact law is used to model the local contact behaviour. Appropriate three-dimensional failure criteria are used for predicting the occurrence of matrix cracking and the extent of delamination after impact.