Time Integration Method Research Articles

Precise eigenvalues in the solutions of generalized Sturm–Liouville problems

In the paper, we determine the eigenvalues and eigenfunctions of the generalized Sturm–Liouville problems, whose potentials may be nonlinear functions of eigen-parameter, by developing new iterative algorithms based on the fictitious time integration method (FTIM) and half-interval method (HIM). We derive two eigen-parameter dependent linear shape functions, from which we can transform the generalized Sturm–Liouville problem to an initial value problem for a new variable, and automatically preserve the prescribed eigen-parameter dependent Sturm–Liouville boundary conditions. A nonlinear equation in terms of the relative norm of two consecutive right-end values of the new variable is derived for iteratively determining the eigenvalue by using the FTIM. The resultant sequence of the iterated eigenvalues are monotonically convergent to the desired eigenvalue, and meanwhile the unknown initial values of the eigenfunction can be determined for computing the eigenfunction by integrating the Sturm–Liouville equation. We propose a high precision point target method by using the HIM to determine the eigenvalue and eigenfunction of the generalized Sturm–Liouville problem, which is transformed to a definite initial value problem with a simpler Dirichlet or Neumann boundary condition on the right-end as a point target equation. Several new theoretical results are proved such that more simple generalized Sturm–Liouville problem can be derived with a single point target on the right-end. Depending on different transformation techniques and the derived target equations, seven types FTIM and four types HIM are developed. Numerical examples confirm that the present FTIM and HIM are effective and accurate.

Time Integration Methods for Nonlinear Dynamic Problems

The accurate simulation and analysis of nonlinear dynamic problems across various scientific disciplines are essential for understanding complex physical phenomena. Time integration methods play a crucial role in numerically solving these problems, providing numerical approximations to the time-evolution of systems governed by ordinary or partial differential equations. This paper presents a comprehensive exploration of time integration methods tailored for addressing nonlinear dynamic problems encountered in diverse fields such as physics, engineering, and computational sciences. The review encompasses a detailed analysis of fundamental time integration techniques, including explicit and implicit schemes, highlighting their respective advantages and limitations in handling nonlinearities. Furthermore, it examines advanced time integration approaches specifically designed to tackle the challenges posed by nonlinear systems. This includes discussions on adaptive time-stepping methods, geometric and symplectic integrators, and other specialized techniques aimed at enhancing accuracy and stability while mitigating computational costs. Through illustrative examples and comparative analyses, this paper evaluates the performance of various time integration methods in addressing nonlinear dynamic problems. It delineates the significance of these methods in accurately capturing system behavior and elucidates their implications for practical applications. Additionally, this review identifies current challenges and outlines prospective directions for further advancements in time integration methodologies tailored to the intricacies of nonlinear dynamic systems. This comprehensive review aims to provide researchers, practitioners, and computational scientists with valuable insights into selecting and implementing appropriate time integration methods for effectively addressing nonlinear dynamic problems across diverse scientific domains. Key Words: Nonlinear Dynamics, Time Integration Techniques, Numerical Simulation, Computational Methods.

Three-dimensional nonlinear vibrations of slightly curved cantilevered pipes conveying fluid

In this paper, nonlinear static and dynamic behaviors of slightly curved cantilevered pipes conveying fluid under transverse base excitations are investigated. Incorporating geometrical and inertial nonlinearities, a new three-dimensional dynamic model is developed for the inextensible cantilevered pipe based on the Euler–Bernoulli beam theory. The resultant partial differential equations are discretized using the Galerkin method and solved utilizing the pseudo-arclength continuation technique and a time-integration method. The static equilibrium configuration, linear stability, and nonlinear response of the pipe are calculated for various cases, which allow us to investigate the effects of flow velocity, gravity coefficient, initial curvature amplitude and shape, and base-excitation amplitude and frequency. An emphasis is put on the importance of modal interactions on frequency-response and force-response relationships, evidencing the possible in-plane and out-of-plane transverse vibrations and identifying the preferred forms of pre- and post-instability behaviors. This study offers an efficient theoretical model for exploring the nonlinear forced response of slightly curved cantilevered pipes conveying fluid and provides a better insight for further investigations on the modal interaction in cantilevered pipe devices.

A Novel Spacetime Boundary-Type Meshless Method for Estimating Aquifer Hydraulic Properties Using Pumping Tests

This article introduces a new boundary-type meshless method designed for solving axisymmetric transient groundwater flow problems, specifically for aquifer tests and estimating hydraulic properties. The method approximates solutions for axisymmetric transient groundwater flow using basis functions that satisfy the governing equation by solving the inverse boundary value problem in the spacetime domain. The effectiveness of this method was demonstrated through validation with the Theis solution, which involves transient flow to a well in an infinite confined aquifer. The study included numerical examples that predicted drawdown at various radial distances and times near pumping wells. Additionally, an iterative scheme, namely, the fictitious time integration method, was employed to iteratively determine the hydraulic properties during the pumping test. The results indicate that this approach yielded highly accurate solutions without relying on the conventional time-marching scheme. Due to its temporal and spatial discretization within the spacetime domain, this method was found to be advantageous for estimating crucial hydraulic properties, such as the transmissivity and storativity of an aquifer.

New options for explicit all Mach number schemes by suitable choice of time integration methods

Many low-Mach or all-Mach number codes are based on space discretizations which in combination with the first order explicit Euler method as time integration would lead to an unstable scheme. In this paper, we investigate how the choice of a suitable explicit time integration method can stabilize these schemes. We restrict ourselves to some old prototypical examples in order to find directions for further research in this field.

Design optimisation of piezoelectric energy harvesters for bridge infrastructure

Vibrational energy harvested from the bridge excitation due to the traffic flow or the wind load can be supplied to sensors in a Structural Health Monitoring (SHM) system and help prolong its service life, reduce chemical battery waste and enable its use in remote locations. A common approach to designing a Piezoelectric Energy Harvester (PEH) consists in tuning its fundamental frequency to some target value (e.g., the fundamental frequency of a bridge). However, such approach does not answer the question if the chosen target frequency is optimal, and does not take into account the possibility to have multiple design configurations with the same fundamental frequency. In this work, we approach the problem of a PEH design optimisation in a rigorous way, using a PEH model based on the Kirchhoff–Love plate theory and Isogeometric Analysis (IGA), coupled with the Modal Order Reduction (MOR) approach and Runge–Kutta time integration method. The model is further equipped with the Particle Swarm Optimisation (PSO) algorithm that allows finding geometry which maximises energy output from a given base acceleration signal. A comprehensive study is conducted to infer the impact of a PEH geometry on its energy harvesting performance in a real-world setting by considering field health monitoring data of a large-scale cable-stayed bridge in NSW, Australia. A shape optimisation framework is developed based on acceleration events, i.e., where the level of response exceeds a certain threshold. Designs obtained in the event-based optimisation are then clustered to propose several best candidates for continuous energy generation. This work represents the first study on PEH design optimisation for real operational conditions.

Combination of Optimal Three-Step Composite Time Integration Method with Multi-Point Iterative Methods for Geometric Nonlinear Structural Dynamics

This study focuses on solving the geometric nonlinear dynamic equations of structures using the multi-point iterative methods within the optimal three-step composite time integration method (OTCTIM). The OTCTIM, initially devised for linear dynamic systems, is now proposed to encompass nonlinear dynamic systems in such a way that the semi-static nonlinear equations in time sub-steps can be solved using multi-point methods. The Weerakoon–Fernando method (WFM), Homeier method (HM), Jarrat method (JM), and Darvishi–Barati method (DBM) have been extended as multi-point solvers for nonlinear equations in OTCTIM, which exhibit a higher convergence order than the Newton–Raphson method (NRM), without requiring the calculation of second and higher derivatives. Several structural examples were solved to examine the performance of these methods in the OTCTIM approach. The results demonstrated that the multi-point iterative methods outperform NRM (in terms of the number of iterations) within the OTCTIM for geometric nonlinear structural dynamics and, among the multi-point methods, the JM and DBM converged with fewer number of iterations and lower error levels. Furthermore, it has been observed that when solving nonlinear dynamic equations for structures with a high number of degrees of freedom, the incorporation of the DBM into the OTCTIM mitigates the convergence iterations and the average elapsed time for iterative sub-steps.

Optimal implicit single-step time integration methods with equivalence to the second-order-type linear multistep methods for structural dynamics: Accuracy analysis based on an analytical framework

This paper presents a comprehensive study of the second-order-type time integration methods family encompassing the linear multistep (LMS) methods, the single-step methods, and the LMS equivalent single-step methods for structural dynamics. An analytical accuracy framework for algorithm design, convergence, and optimization is developed, where the obstacles encountered in algorithms construction, convergence accuracy, algorithms equivalence, accuracy measurement, and accuracy optimization in traditional accuracy frameworks are well addressed. Rigorous accuracy analysis successively to the second-order-type LMS (LMS2) methods, the single-step methods, and the LMS2 equivalent single-step methods aims to reveal optimal algorithms with desirable numerical properties. First, independent implicit algorithms including the velocity type and the acceleration type LMS2 methods except the displacement type ones are newly revealed in the LMS2 methods family with second-order convergence and A-stability, and three error constants that can play essential roles in the accuracy measurement and algorithm optimization are developed. Second, a single-step methods family where the displacement, velocity, and acceleration can achieve the same order of accuracy simultaneously is revealed in a generalized 16-parameters methods family, which contains numerous conventional single-step single-solve methods as subfamily methods. It is found that the unit consistent and non-consistent representations show a significant difference in the local truncation error, stability, and convergence performances. Third, a family of accuracy-improved single-step methods with algorithmic equivalence, which is stricter than the traditional spectral equivalence, to the LMS2 methods is revealed. It is clarified that the difference between the methods with equivalence can significantly influence their convergence accuracy, and the error constants concept of the LMS2 methods can be extended to the single-step method with algorithmic equivalence. Last, an algorithm optimization strategy is developed based on the error constants and the ultimate spectral radius, and an optimal single-step method with a single parameter to control the accuracy and numerical dissipation is revealed analytically without empirical assumptions. The optimal method is second-order accurate, unconditionally stable, overshooting acceptable, and equivalent to an optimal route between the Houbolt method and the trapezoidal rule, and its numerical performance is verified by the spectral accuracy and two numerical examples.

A Boundary-Type Numerical Procedure to Solve Nonlinear Nonhomogeneous Backward-in-Time Heat Conduction Equations

In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique.

Resolving entropy growth from iterative methods

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.

Functional‐preserving predictor‐corrector multiderivative schemes

AbstractIn this work, we develop a class of high‐order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite‐Birkhoff‐Predictor‐Corrector (HBPC) methods and the technique of relaxation for numerical methods of ordinary differential equations (ODEs). We explain the algorithm in detail and show numerical results for two‐ and three‐derivative methods, comparing relaxed and unrelaxed methods. The numerical results demonstrate that, at the slight cost of the relaxation, an improved scheme is obtained.

Implicit–explicit multirate infinitesimal stage-restart methods

Implicit–Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is split into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems split into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit–explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further split in an IMEX fashion. This new class, which we call implicit–explicit multirate infinitesimal stage-restart (IMEX-MRI-SR), both improves upon the previous implicit–explicit multirate infinitesimal generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods by allowing for far easier creation of new embedded methods, and extends multirate exponential Runge Kutta (MERK) methods by allowing the fast-changing dynamics to be nonlinear and the methods to be implicit. We leverage GARK theory to derive conditions for orders of accuracy up to four, and we provide second- and third-order accurate example methods, which are the first known embedded MRI methods with IMEX structure. We then perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step settings.

A machine learning-based hybrid seismic analysis of a lead rubber bearing isolated building specimen

This paper presents the first experimental validation of the hybrid seismic analysis, previously developed by the authors, by simulating the structural behavior of a full-scale base-isolated building specimen tested at the E-Defense. The two-dimensional structural behavior of the lead rubber bearings (LRBs) is predicted by a machine learning model (MLM) previously developed by the authors, whereas analytical models describe the response of the superstructure. A Python script has been developed to incorporate both analytical models and MLMs into an explicit time integration method to perform three-dimensional hybrid seismic analyses. Moreover, conventional time history analyses have been carried out using a bilinear model for the isolation devices. Experimental, ML-based, and conventional analysis responses of 14 shake table motions are processed and compared in terms of hysteresis curves of the isolation layer, roof displacement and peak floor acceleration. Overall, hybrid seismic analysis showed better results than conventional one in terms of capturing the energy dissipation capacity of the isolation layer (maximum a20-index of 0.859 for the drift and 0.720 for the shear with respect to experiment). Similarly, roof displacement was predicted with a maximum a20-index value of 0.832. Besides, the maximum average computation time of hybrid seismic analysis was 1.78 s per integration time-step. Therefore, the efficiency of the hybrid seismic analysis is experimentally proven in this paper in terms of both accuracy and applicability. This machine learning-based analysis method is strongly recommended to simulate time-history response of complex structures.

Rosenbrock-W methods for stochastic Galerkin systems

The stochastic Galerkin approach is a widely used method for quantifying uncertainty in dynamical systems. A polynomial chaos expansion of physical variables is used to represent uncertainty. Thus the evolution equations for the dynamics of physical quantities are replaced by evolution equations for the polynomial chaos coefficients. The dimension of the stochastic Galerkin system is m times larger than the dimension of the original physical system, where m is the number of polynomial chaos coefficients. The cost of implicit time integration methods, required for stiff dynamical systems, increases considerably, for example, by a factor m3 when direct linear solvers are used.This work studies efficient implicit methods for the time integration of stiff stochastic Galerkin systems consisting of ordinary differential equations. Linearly-implicit Rosenbrock–Wanner schemes are considered. A block-diagonal approximation of the Jacobian of the stochastic Galerkin system is employed, where the diagonal blocks are the Jacobian of the physical system, evaluated at the mean solution value. We perform a theoretical study of numerical stability for this approximation, using a stochastic extension of the Dahlquist test equation, and propose a concept of linear stochastic Galerkin stability. A rigorous analysis is done for methods of order one and two. A third-order scheme is designed to investigate the stability properties by numerical experiments. Numerical results confirm the theoretical findings.

Application of PGD separation of space to create a reduced-order model of a lithium-ion cell structure

Lithium-ion cells can be considered a laminate of thin plies comprising the anode, separator, and cathode. Lithium-ion cells are vulnerable toward out-of-plane loading. When simulating such structures under out-of-plane mechanical loads, subordinate approaches such as shells or plates are sub-optimal because they are blind toward out-of-plane strains and stresses. On the other hand, the use of solid elements leads to limitations in terms of computational efficiency independent of the time integration method. In this paper, the bottlenecks of both (implicit and explicit) methods are discussed, and an alternative approach is shown. Proper generalized decomposition (PGD) is used for this purpose. This computational method makes it possible to divide the problem into the characteristic in-plane and out-of-plane behaviors. The separation of space achieved with this method is demonstrated on a static linearized problem of a lithium-ion cell structure. The results are compared with conventional solution approaches. Moreover, an in-plane/out-of-plane separated representation is also built using proper orthogonal decomposition (POD). This simply serves to compare the in-plane and out-of-plane behaviors estimated by the PGD and does not allow computational advantages relative to conventional techniques. Finally, the time savings and the resulting deviations are discussed.

A novel single-step explicit time integration method based on momentum corrector technique for structural dynamic analysis

In this study, a novel single-step explicit time integration method based on momentum corrector technique is proposed. The algorithmic procedure is formulated using cubic B-spline interpolation and Taylor series expansion. The momentum corrector is introduced to enhance solution accuracy of problems, especially for discontinuous load problems. Through the theoretical analysis of local truncation errors, spectral radius, algorithmic damping and period elongation, several parameter cases are obtained to achieve enhanced accuracy and stability. Numerical examples, including a standard single-degree-of-freedom problem, two discontinuous load problems, a seismic response analysis problem, a wave propagation problem and a nonlinear problem, are tested. The results illustrate the new method achieves high level of accuracy for discontinuous load problems, while simultaneously exhibiting significant accuracy improvement when dealing with other problems.

Dispersion analysis of numerical schemes using 2D compressible linearized Navier–Stokes equation for direct numerical simulation

The dispersion analysis of numerical schemes for finite difference computations based on 2D compressible linearized Navier–Stokes equation (LNSE) is presented here. The analysis presents a more holistic view of the applicability of the schemes while solving compressible NSE, rather than analyzing model 1D or 2D convection or convection–diffusion equations. A theoretical dispersion analysis of model 2D−LNSE is performed first which categorizes it into vortical, entropic and two acoustic modes. The variation of the theoretical dispersion relation for each of the modes with wavenumber, Mach number and Reynolds number is analyzed. It is noted that the entropic mode is the most diffusive among all the modes. Two acoustic modes are less diffusive than the vortical mode up to a certain absolute wavenumber Kb depending upon the Mach number and Reynolds number. It is also shown that while vortical and entropic modes are non-dispersive in nature, the two acoustic modes are dispersive only up to K=Kb. Next, numerical dispersion and diffusion corresponding to each of the modes are analyzed for second order central difference (CD2) and sixth order central compact difference scheme proposed in Lele (J. Comput. Phys., Vol-103(1), 1992) for discretization of convective and diffusive derivatives using fourth-order Runge–Kutta (RK4) time integration method. Results show the existence of numerical unstable zone and regions where spurious upstream propagating waves are noted for each of the modes in the spectral plane. From these and the relative size of the dispersion relation preserving zones for the modes, applicability of the numerical schemes can be assessed. Analysis of the variation of the numerical unstable zone with Mach number, orientation of the mean flow and the unit CFL number Nc=Δt/Δx for each of the modes are presented. It is noted that entropic mode displays the lowest critical CFL number of all the four modes. Lastly, linearized and nonlinear numerical simulations for the convection of acoustic and vortical pulse with varying spectral content are presented. The linearized simulations are also compared with corresponding exact solutions. Obtained results indicate a direct correspondence of the stability analysis performed based on 2D−LNSE even for nonlinear simulations which result in additional intra modal energy transfer and interactions.

Performance of the quadrilateral overlapping elements combined with implicit time integration method in solving the transient wave propagations

In the analyses of transient wave propagations, a numerical method with the monotonic convergence property is favored by analysts. However, the classic finite element method does not possess this attractive property due to pronounced numerical dispersions and can generate severe spurious oscillations by using a small time step. In this work, the novel quadrilateral overlapping element is extended into the investigation of transient wave propagations by using the standard implicit Bathe time integration method. Since the local field approximation in overlapping elements can be a function, the performances of different basis functions in transient wave propagation problems are studied. Detailed analyses regarding the numerical dispersion error caused by spatial and temporal discretization are conducted. Then several numerical experiments with different types of excitation loading are investigated to show the performance of the quadrilateral overlapping elements. It is shown that the overlapping element with the trigonometric functions and the bilinear Lagrange polynomials has the monotonic convergence property and can perform well in handling various transient wave propagation problems.

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.