Direct Time Integration Research Articles

A novel iterative algorithm for nonlinear inelastic dynamic analysis of truss structures; MCB-Spline+Sp time integration method

The focus of this work is on the development of a novel algorithm for the nonlinear dynamic analysis of structures. In the sake of a general and easy-to-implement solution method, the tangential stiffness matrix is calculated by combining the Cauchy-Riemann equations derived from the differentiation of complex functions and using the nodal displacement perturbation vector. An implicit direct time integration method based on a cubic B-Spline is integrated with the incremental-iterative method using a quadrature rule based on the interpolation of spline functions, which is referred to as the MCB-Spline+Sp time integration method. A step-by-step algorithm for developing a computer code to implement this method is provided. The effectiveness of the proposed approach is evaluated through several nonlinear time history analyses, which demonstrate its accuracy and efficiency by observing less computational efforts and fast convergence rate. Moreover, a comparison is made between the developed method and well-established techniques such as the Newmark and Standard-Bathe time integration methods in combination with the iterative Newton-Raphson method.

The α-generalized implicit method associated with Potra–Pták iteration for solving non-linear dynamic problems

Generally, direct time integration procedures are used for solving the equations of motion in transient analysis of structures with large displacements. In this context, we propose an algorithm that combines the α-Generalized implicit integration method with the Potra–Pták two-step iterative scheme. The free Scilab program develops a computer code for the non-linear dynamic analysis of plane frames with large displacements and rotations. The FEM corotational formulation discretizes the structures considering the Euler-Bernoulli beam theory. The null-length connection element described by the axial, translational and rotational stiffnesses simulate the behavior of the beam-column connection. Jacobi’s method and the Scilab’s spec function determine the natural frequencies. The developed program is used for modal and transient dynamic analyses of frame problems available in the literature. The numerical results show that the Potra-Pták scheme obtains approximate solutions with fewer cumulative iterations until convergence and shorter processing time compared to the standard Newton-Raphson scheme. Further-more, the results show that the type of beam-column connection affects the vibratory behavior of the structure as well as the values of its natural frequencies.

OPTIMIZING LEARNING: A META-ANALYSIS OF TIME MANAGEMENT STRATEGIES IN UNIVERSITY EDUCATION

This meta-analysis rigorously investigates time management strategies within university education and their impact on academic performance, student satisfaction, and self-efficacy. Covering two decades (2004-2024), it synthesizes findings from both random and nonrandom controlled intervention studies, providing a comprehensive examination of pedagogical impacts. The analysis reveals nuanced but significant enhancements across academic and personal development dimensions, with varied effect sizes (average g values). Interventions were categorized based on their primary focus—either direct time management skills enhancement or integration within broader academic and personal development strategies. Directly targeting time management proficiency was pivotal in bolstering academic performance and self-efficacy, while broader interventions significantly elevated student satisfaction. This analysis highlights the critical interplay between time management strategies, self-efficacy, and satisfaction, suggesting that tailored, skill-specific interventions can yield substantial benefits in academic contexts. The findings advocate for a holistic approach in higher education pedagogy, emphasizing the necessity of embedding effective time management training within university curricula to optimize learning outcomes and student development. This study contributes to the academic discourse on educational strategies and underscores the imperative for empirical, evidence-based approaches in curriculum design and student support services, guiding future research in educational psychology.

Wave packet enriched finite element for accurate modelling of thermoelectroelastic wave propagation in functionally graded solids

A wave packet enriched finite element (FE) formulation featuring electrothermomechanical coupling is presented for solving two-dimensional wave propagation problems in functionally graded elastic and piezoelectric media. The FE equations for the Lord-Shulman and Green-Lindsay theories of generalized piezothermoelasticity are derived, where the standard Lagrangian interpolation functions of the temperature, electric potential, and displacement field variables are extrinsically enriched with element-domain sinusoidal wave-packet functions and are solved using the Newmark–β direct time integration. The effective properties of the functionally graded material (FGM) are computed using the volume-fraction-based micromechanics models, the Mori-Tanaka model and Voigt's rule of mixture (ROM) model. A variety of wave propagation problems, including the mechanically and electrically excited high-frequency Lamb wave propagation in FGM plates, impact waves in FGM bars, and thermal shock waves in functionally graded piezoelectric cylinders, are solved using the present formulation, and results are validated with available solutions. The efficacy of the current solution in comparison with the conventional FE is evaluated for all the studied problems. The effect of the inhomogeneity parameter on the transient wave behaviour is also presented for all the problems. The results of the Mori–Tanaka model are compared with those of the widely used ROM.

A practical way to apply a technique that accelerates time history analysis of structures under digitised excitations

Time history analysis using direct time integration is a versatile and widely accepted tool for analysing the dynamic behaviour of structures. In 2008, a technique was proposed to accelerate thetime history analysis of structural systems subjected to digitised excitations. Recently, this techniquehas been named as the SEB THAAT* (Step-Enlargement-Based Time-History-Analysis-Acceleration-Technique), and the determination of appropriate values for its parameter is introduced as the mainchallenge. To overcome this challenge, a procedure is proposed in this paper. The basis of the procedure is the comments on accuracy control in structural dynamics and numerical analysis ofordinary differential equations, legalised in the New Zealand Seismic Code, NZS 1170.5:2004. As themain achievement, by using the proposed procedure, we can apply the SEB THAAT and carry out thetime history analysis clearly and with less parameter setting compared to the ordinary time historyanalysis. The proposed procedure is always applicable and, except when the behaviour is very complex, oscillatory and non-linear, the reductions in analysis run-time are considerable while the changes in accuracy are negligible. The performance can be sensitive to the problem, the integration method, the target response, and the severity of the non-linear behaviour. Compared to the previous tests on the SEB THAAT, the efficiency of applying the SEB THAAT using the proposed procedure is better, the sensitivity of the performance to the problem is lower, and a measure of accuracy is available. Compared to other techniques for accelerating structural dynamic analyses, the use of the SEB THAAT according to the proposed procedure has several positive points, including the simplicity of implementation.

Enhancing predictive capabilities in data-driven dynamical modeling with automatic differentiation: Koopman and neural ODE approaches.

Data-driven approximations of the Koopman operator are promising for predicting the time evolution of systems characterized by complex dynamics. Among these methods, the approach known as extended dynamic mode decomposition with dictionary learning (EDMD-DL) has garnered significant attention. Here, we present a modification of EDMD-DL that concurrently determines both the dictionary of observables and the corresponding approximation of the Koopman operator. This innovation leverages automatic differentiation to facilitate gradient descent computations through the pseudoinverse. We also address the performance of several alternative methodologies. We assess a "pure" Koopman approach, which involves the direct time-integration of a linear, high-dimensional system governing the dynamics within the space of observables. Additionally, we explore a modified approach where the system alternates between spaces of states and observables at each time step-this approach no longer satisfies the linearity of the true Koopman operator representation. For further comparisons, we also apply a state-space approach (neural ordinary differential equations). We consider systems encompassing two- and three-dimensional ordinary differential equation systems featuring steady, oscillatory, and chaotic attractors, as well as partial differential equations exhibiting increasingly complex and intricate behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the state-space approach offers superior performance compared to the "pure" Koopman approach where the entire time evolution occurs in the space of observables. When the temporal evolution of the Koopman approach alternates between states and observables at each time step, however, its predictions become comparable to those of the state-space approach.

Implicit direct time integration of the heat conduction problem in the Method of Matched Sections

The paper is devoted to further elaboration of the Method of Matched Sections as a new branch of finite element method in application to the transient 2D temperature problem. The main distinction of MMS from conventional FEM consist in that the conjugation is provided between the adjacent sections rather than in the nodes of the elements. Important feature is that method is based on approximate strong form solution of the governing differential equations called here as the Connection equations. It is assumed that for each small rectangular element the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, , and heat flux . In practical realization for rectangular finite elements the method is reduced to determination of eight unknowns for each element – two unknowns on each side, which are related by the Connection equations, and requirement of the temperature continuity at the center of element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step became the parameter of shape function within the element, i.e. it determines the behavior of the Connection equations. This method was early proposed by first author for number of 1D problem, and here in first time it is applied for 2D problems. The number of tests for rectangular plate exhibits the remarkable properties of this “embedded” time integration scheme with respect to stability, accuracy, and absence of any restrictions as to increasing of the time step.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

The effect of an explosive shock wave on the plate of a protective structure of a critical infrastructure

The article presents the outcomes of an analysis on the stress-strain conditions of reinforced concrete structures subjected to an explosive shock wave resulting from the detonation of a combat unit from a kamikaze drone against a protective screen. When designing protective structures for critical infrastructure, employing computer simulation enables an assessment of the genuine impact of explosive loading on the structural elements' strength. The active phase of explosive loading is exceptionally brief, lasting only a fraction of a second. Under such circumstances, modeling is best performed using explicit methods of direct integration in time.
 The structure considered in this work is a reinforced concrete slab supported by a metal beam cage with I-beam cross-sections, topped with a sand backfill. The study was executed within the SIMULIA Abaqus software suite, incorporating models depicting nonlinear material behavior in a three-dimensional context. Discretely positioned reinforcement was considered for reinforced concrete structures, and the "Concrete Damage Plasticity" model was applied for concrete, accounting for damage accumulation. The devised computational scheme for the shelter represents a section of the protective structure's roof under conditions of cyclic symmetry.
 The article elucidates the core principles of incorporating explosive loading according to algorithm CONWEP.The results demonstrate that during the detonation of a kamikaze drone, an explosive wave created a crater in the sand backfill, exposing the slab. The study illustrates the development of damage in the reinforced concrete slab at various time intervals. Despite the identified damage to the slab, the protective structure overall withstood the explosive load. The intensity of the explosive shock wave diminishes significantly as it propagates away from the explosion site.

Multiple regimes of lock-in and hysteresis in free vibration of a rotating cylinder

Flow-induced vibration of a rotating cylinder that is free to oscillate in the stream-wise and cross stream directions is studied in the laminar flow regime via linear stability analysis (LSA) and direct time integration (DTI). LSA reveals that the instability can arise from fluid-, elastic-, or coupled fluid-elastic modes depending on the rotation rate of the cylinder, α, and reduced speed, U*. Beyond α=2, the U*-range of lock-in increases exponentially with an increase in α. DTI brings out the multiple regimes of lock-in at various α. Each lock-in regime extends for a certain range of U* and is associated with a different mode of vortex shedding. The modes differ in terms of the number of pair of vortices shed during each cycle of cylinder oscillation. The amplitude of cylinder oscillation increases with an increase in the number of shed vortices. With an increase in Re, the number of vortex shedding modes increase. Vortices are generally shed during the upstream movement of cylinder, while the shear layer wraps around it resulting in large lift during the downstream motion. The flow as well as the oscillation amplitude is found to be sensitive to the initial condition for a certain range of α and U*. A flow regime is discovered where three distinct response states can be realized depending on the initial condition. Hysteresis in response to the cylinder and flow, with respect to increase and decrease in U*, occurs near the transition between lock-in and desynchronization and during the switch in the mode of vortex shedding.

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

An Equivalent Linearization technique, termed an Equivalent Linearization Time and Ensemble Expectation (EL-TEE) approach, is used to develop an alternative method for estimating the response of a nonlinear oscillator to a combination of deterministic harmonic and random white noise excitation. The approach is based on applying equivalent linearization and averaging over the time period of one harmonic excitation cycle. This gives a set of coupled nonlinear equations that can be solved for the response averaged over time and across the ensemble. The primary advantages of the proposed method are its computational speed, ability to return physically meaningful linearization matrices and that it can be applied to a wide variety of nonlinearities. The method is applied to three example test systems: the well-known single degree of freedom Duffing oscillator; a single degree of freedom system with a displacement constraint imposing a discontinuous nonlinearity; and a multi degree of freedom oscillator with a localized polynomial nonlinearity that has also been examined experimentally. It is shown that the response predicted matches well with Monte Carlo results from direct time integration at a fraction of the computational cost, and the method is capable of reproducing key results observed experimentally.

Asynchronous time integration while achieving zero interface energy

This contribution deals with an asynchronous direct time integration of the finite-element model. The proposed method is applied to the phenomenon of wave propagation through an elastic linear continuum. The numerical model is partitioned into individual subdomains using the domain decomposition method by means of localized Lagrange multipliers. For each subdomain, different time discretizations are used. No restrictions for relation between subdomain’s time steps are imposed. The coupling of the subdomains is forced by an acceleration continuity condition. Additionally, we use the a posteriori technique to also provide the displacement and velocity continuity at the interfaces, and hence we obtain exact continuity of all three kinematic fields. The proposed method is experimentally validated using the modified SHPB (split Hopkinson pressure bar) setup.

Spatial Variation of the Coefficient of Restitution for Frictionless Impacts on Circular Beams

Abstract The spatial variation of the coefficient of restitution for frictionless impacts along the length of a circular beam is investigated using a continuous impact model. The equations of motion are obtained using the finite element method, and direct time integration is used to simulate the collision on a fast time scale. For collision of a pinned beam with a fixed cylinder, the spatial variation of the coefficient of restitution, impulse magnitude, duration of collision, energetics, and the role of damping are investigated. In the absence of significant external damping, the kinematic and kinetic definitions of the coefficient of restitution provide identical results. Experiments validate the results from simulation which indicate that the coefficient of restitution is sensitive to the location of impact.

On designing and developing single‐step second‐order implicit methods with dissipation control and zero‐order overshoots via subsidiary variables

AbstractHilber and Hughes gave several competitive demands on the implicit methods in 1978. However, there are no such integration methods in the traditional algorithm design. Via imposing subsidiary variables, this article proposes two novel implicit methods to splendidly achieve these competitive demands without increasing computational costs. The two novel methods are self‐starting, unconditionally stable, single‐solve, identically second‐order accurate, controllably dissipative, and non‐overshooting. The novel methods achieve the maximal dissipation of auxiliary variables in the high‐frequency limit, although such the design is not a must. The first novel method uses auxiliary velocity and acceleration variables, whereas the second one employs two auxiliary acceleration variables. After embedding desired numerical characteristics, the novel methods employ four eigenvalues of amplification matrices in the high‐frequency limit as user‐specified parameters to flexibly control numerical high‐frequency dissipation. To reduce the number of use‐specified parameters, this article gives some recommended sub‐families of algorithms, such as optimal low‐frequency dissipation schemes. The article further refines the error analysis to analytically calculate amplitude and phase errors for some implicit methods. The error analysis technique can be applied to almost all direct time integration methods. Comparisons of amplitude and phase errors highlight the superiority of the novel methods over the published algorithms. In addition, some classical measures, such as numerical damping ratios and relative period errors, are analyzed and compared. Numerical examples are solved to show the novel methods' superiority.

Quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass

In this work, quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass are studied. Coupled modelling equations of nonlinear vibration of an axially moving beam are established with the aid of the generalized Hamilton principle. A nonlinear partial-integro-differential equation as the transverse vibration modelling equation is deduce by decoupled method. After obtaining the equilibrium buckled state of the axially moving beam, a transverse vibration governing equation around the buckled state is presented. The Galerkin method is applied to truncate the governing equation into a set of nonlinear ordinary differential equations. The harmonic balance method with three time variables is applied to analyze quasi-periodic vibration of the nonlinear dynamic system with combined parametric and forced external excitation. Time traces and phase-plane portraits of quasi-periodic vibration are obtained, which are in excellent agreement with those of the direct time integration method. Frequency amplitude responses of the vibration are determined by the harmonic balance method with three time variables. Effects of the velocity of the axially moving beam, amplitude of the varying mass, and the external excitation frequency on the vibration amplitude of axially moving beam and the ratio of the two fundamental frequencies are investigated.

The edge-based smoothed FEM with [formula omitted]-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel ρ∞-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and ρ∞-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel ρ∞-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

Energy-conserving interface dynamics with asynchronous direct time integration employing arbitrary time steps

The derivation and implementation of an asynchronous direct time integration scheme for domain-decomposed finite element models is presented. To maximize clarity in the description of the proposed asynchronous integration, the scheme is restricted to the linear-elastic stress wave propagation case. The proposed method allows the integration of individual subdomains with independent time steps. There is no requirement for an integer time steps ratio of the interacting domains while maintaining zero interface energy. The subdomains are connected by the condition of the continuity of the acceleration field at the interface. In addition, the a posteriori technique is applied to satisfy the continuity of the displacement and velocity fields. Another important contribution of this paper lies in the description of the implementation — we offer the reader a general description of the implementation of the case of any number of subdomains with any number of constraints between them, while the basics of the algorithm are explained on a single domain pair. The functionality of the asynchronous integrator is verified by solving selected problems and comparing with analytical solutions and experimental measurements obtained using a Split Hopkinson pressure bar setup.

Wave packet enriched finite elements for thermo-electro-elastic wave propagation problems with discontinuous wavefronts

In this article, we study the numerical properties of the recently developed local (element-domain) harmonic basis function enriched finite element (FE) for wave propagation problems. The study includes the effect of the enrichment harmonics on the conditioning of system matrices and how the diagonal scaling mass lumping technique influences the possible ill-conditioning and the solution accuracy. Subsequently, a wave packet enriched thermoelectromechanical FE formulation is presented for axisymmetric and planar wave propagation problems in piezoelastic media. The conventional Lagrange interpolations for the displacement, temperature, and electric potential fields are enriched with the element-domain sinusoidal functions which satisfy the partition of unity condition. The extended Hamilton’s principle and constitutive relations of the Lord–Shulman and Green–Lindsay generalized piezothermoelasticity are employed to derive the coupled system of equations of motion which is solved using Newmark- direct time integration scheme. The performance of the proposed elements is assessed and wave characteristics are studied for the problems of thermoelastic shock waves in an elastic annular disk and thermoelectric shock waves in a piezoelectric hollow cylinder. The element shows significant improvement in the computational efficiency and accuracy over the conventional FE for these problems involving sharp discontinuities in the fields at the wavefronts.

Semi-analytical implicit direct time integration method for 1-D gas dynamic problem

Sharp wave treatment for 1-D gas dynamic problem is still a chellenge for modern numerical methods. They often require too many space and time steps, produce spurious oscillation of solution, exhibit a strong numerical dissipation or divergence of results. This paper is further extension of authors’ idea of employment the analytical solution for space coordinate, where time step is a parameter which used in the space solution. Its peculiarity consists in development of additional procedure of linearization of dependence between the pressure and density. It is performed in premise that actual pressure for each space element is close to the basic pressure, attained at previous moment of time. The efficiency of method is tested on the very popular task of Sod, where two different ideal gases in a tube are separated by diaphragm, which is suddenly broken. The problem considered in Lagrangian coordinates formulation. The results obtained show the very good efficiency of method, which requires the essentially lesser time and space steps, leads to no spurious oscillation and give consistent and predictable results with respect to meshing. The accuracy of method is mostly controlled by time step, which should be larger than clearly stated theoretical lower limit. Other advantage of method is that it can calculate the process to any desired moment of time, and space meshing can be variable in time and space and can be easily adapted during the process of calculation.

Operational Absolutely Optimal Dynamic Control of the Stochastic Differential Plant’s State by Its Output

The problem of synthesizing the average-optimal control law for a dynamic plant subject to random disturbances, if its state variables are measured partially or with random errors, is considered. Using the method of a posteriori sufficient coordinates (SCs), the complexity of constructing the well-known interval-optimal Mortensen controller is described and a much simpler algorithm for finding its operational-optimal analog is obtained. The new controller does not require the solution of the corresponding Bellman equation in inverse time, since it is optimal in the sense of a time-varying criterion. This makes it possible to disregard information about the future behavior of the object and reduces the procedure for finding the dependence of a control on sufficient coordinates to direct-time integration of the Fokker–Planck–Kolmogorov equation and to solving a problem of parametric nonlinear programming. The application of the obtained algorithm is demonstrated by the example of a linear-quadratic-Gaussian problem, as a result of which a new operational version of the well-known separation theorem is formulated. It represents a stochastic control device as a combination of a linear Kalman–Bucy filter and a linear operational-optimal positional controller. The latter differs from the traditional interval-optimal controller by the well-known gain and does not require the solution of the corresponding matrix Riccati equation in inverse time