Time Integration Method Research Articles

Improved local time-stepping schemes for storm surge modeling on unstructured grids

This paper presents improved explicit local time-stepping (LTS) schemes of both first and second order accuracy for storm surge modeling. The two-dimensional shallow water equations are numerically solved on unstructured triangular meshes using finite volume method with Roe’s approximate Riemann solver. The LTS algorithms are designed based on explicit Euler and strong stability preserving Runge–Kutta time integration methods. A single-layer interface prediction–correction scheme is adopted to combine coarse and fine time discretization, further enhancing the stability of the LTS schemes, particularly at higher LTS levels and during long time simulations. An ideal numerical test validates the efficiency of the improved LTS models, revealing their capability to improve computational speed while preserving conservation properties and reducing accuracy loss as LTS levels increase. We further apply the LTS models to cross-scale simulations of storm surges in the Northwest Pacific. Results show that compared to the global time-stepping (GTS) models, the LTS models significantly boost computational speed by up to 37%, all while delivering equally reliable computational outcomes. With expanding high-resolution coastal data and the need for high-resolution modeling, the improved LTS models show great potential for cross-scale storm surge modeling.

Tsunami Squares: Leapfrog scheme implementation and benchmark study on wave–shore interaction of solitary waves

Impulse waves are generated by rapid subaerial mass movements including landslides, avalanches and glacier break-offs, which pose a potential risk to public facilities and residents along the shore of natural lakes or engineered reservoirs. Therefore, the prediction and assessment of impulse waves are of considerable importance to practical engineering. Tsunami Squares, as a meshless numerical method based on a hybrid Eulerian–Lagrangian algorithm, have focused on the simulation of landslide-generated impulse waves. An updated numerical scheme referred to as Tsunami Squares Leapfrog, was developed which contains a new smooth function able to achieve space and time convergence tests as well as the Leapfrog time integration method enabling second-order accuracy. The updated scheme shows improved performance due to a lower wave decay rate per unit propagation distance compared to the original implementation of Tsunami Squares. A systematic benchmark testing of the updated scheme was conducted by simulating the run-up, reflection and overland flow of solitary waves along a slope for various initial wave amplitudes, water depths and slope angles. For run-up, the updated scheme shows good performance when the initial relative wave amplitude is smaller than 0.4. Otherwise, the model tends to underestimate the run-up height for mild slopes, while an overestimation is observed for steeper slopes. With respect to overland flow, the prediction error of the maximum flow height can be limited to ± 50% within a 90% confidence interval. However, the prediction of the front propagation velocity can only be controlled to ± 100% within a 90% confidence interval. Furthermore, a sensitivity analysis of the dynamic friction coefficient of water was performed and a suggested range from 0.01 to 0.1 was given for reference.

An Explicit Time Integration Method Based on the Verlet Scheme with Improved Characteristics in Numerical Dispersion

An Explicit Time Integration Method Based on the Verlet Scheme with Improved Characteristics in Numerical Dispersion

Effect of Sinusoidal Fiber Waviness on Non-Linear Dynamic Performance of Laminated Composite Plates with Variable Fiber Spacing

This study investigated influence of varying waviness characteristics of fiber, represented by path amplitude Δ and different numbers of half sine waves k , on the elastic-plastic dynamic behaviour of laminated composite plates with variable fiber spacing. The analysis was based on the equations for action of constant axial dynamic loading and two-dimensional layered approach with classical first order shear deformation theory with five degrees of freedom per node, and it was performed with FORTRAN 94 programming language. Von-Karman’s assumptions were used for the discretization of the laminated plates to include geometric nonlinearity for nine-node Lagrangian isoperimetric quadrilateral elements. Complete bond between the layers was assumed with no delamination, which was based on first-order shear deformation theory. The Newmark implicit time integration method and Newton-Raphson iteration were simultaneously used to solve the nonlinear governing equation in conjunction. It was proven in the research that the nonlinear performance of the laminated composite plate was affected by the studied waviness parameters Δ and k , and also by the variable distribution pattern selected for this study.

Towards optimal use of the explicit [formula omitted]-Bathe time integration method for linear and nonlinear dynamics

In an earlier publication, we proposed a new explicit time integration scheme, the β1/β2-Bathe method, which is simple in its formulation and showed remarkable accuracy in the solution of problems [1]. A particular strength of the method is that it can directly be used as a first-order or second-order scheme by a change of the values of β1 and β2. While good results are obtained with reasonable values of β1 and β2, for excellent accuracy better values of the parameters need to be chosen. We propose in this paper values of β1 and β2 for the first-order scheme, best used in wave propagation analyses, and separate values for the second-order scheme, best used in analyses of structural vibrations. In each case, one set of values of (β1,β2) is given and to possibly improve the results only one of the parameters needs to be changed, that is, β1 for wave propagations and β2 for structural vibrations, making the scheme a one-parameter method. Another strength of the procedure is that physical damping can directly be included in the solution, the effect of which on the stability and accuracy of the solutions we analyze in the paper. The use of the solution scheme in nonlinear analysis is, as we show in the paper, a simple extension from linear analysis. Finally, we give various solutions using the explicit β1/β2-Bathe method in linear and nonlinear analyses to illustrate the performance of the method with the given recommendations for its use.

Non-smooth dynamics of impacting viscoelastic pipes conveying pulsatile fluid

This paper aims to investigate the stability and nonlinear dynamics of an initially curved viscoelastic clamped–clamped pipe conveying pulsatile fluid, with special attention to the non-smooth dynamics of this gyroscopic system under the influence of initial curvature and gravity. In this study, the impacting contact is modeled as a trilinear repulsive force. Based on the Euler–Bernoulli beam theory, the nonlinear governing equations, which take into account geometric, curvature, and damping nonlinearities, are developed using Hamilton’s principle. The continuous model is discretized using the Galerkin method and then solved using the arc-length technique and a time integration method. A linear analysis of the natural frequency and complex mode shape shows the unexpected frequency intersection and modal asymmetry. The qualitative changes in the nonlinear dynamics of the impacting pipe in both sub- and super-critical regions are examined and presented in the form of bifurcation diagrams, frequency–amplitude curves, force–amplitude curves, time histories, and phase portraits. Numerical results show that some complex dynamical behaviors can occur in different impact constraint cases, including a grazing bifurcation, sticking, period doubling, and chaos at the impact location, may occur. The initial curvature and gravity result in different behaviors, such as the asymmetry change of the equilibrium configuration and the occurrence of the flutter instability. The current analysis is significant to understand the non-smooth dynamic mechanisms and also provide design and technology guidance for the pipe system with motion-limiting constraints.

Inerter-based tuned mass dampers for response control of multi-degree-of-freedom structures subjected to wind and earthquake excitations

The present work explores the utilization of the tuned mass damper-inerter (TMDI) in controlling the structural response subjected to wind and earthquake excitations. An example of a 25-storey reinforced concrete building is taken up. The equations of motion of the structure-TMDI combined system are established by assuming the building behaves as a shear-building with multiple degrees of freedom. These equations are solved using Newmark’s time integration method. The objective functions are the wind- and earthquake-induced structural response quantities, such as the maximum inter-storey drift ratio (MIDR) and peak floor acceleration (PFA). The prime contribution of this study is the inclusion of statistical measures such as the mean, 85th, and 95th percentiles for these quantities. A genetic algorithm-based multi-objective optimization is performed to estimate the optimum TMDI parameters. Based on the outcomes of this study, it can be recommended that the TMDI be optimized to perform well under the 95th percentile for MIDR and PFA in effectively reducing the peak structural responses. Frequency domain analysis results demonstrate TMDI’s multiple-mode controlling capability. The energy comparison plots of the uncontrolled and controlled structures prove the effectiveness of the optimum TMDI in providing enough damping and ensuring structural stability.

Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians

Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.

Study of pulse wave phenomena associated with blood flow model in human viscoelastic artery

Cardiac output monitoring has proven to be a promising hemodynamic management tool especially for critically ill patients. Pulse wave analysis is a noninvasive method used to quantify cardiac output continuously with respect to time. In this article, we have proposed a novel methodology to quantify the contribution of pulse waves to further study the role of arterial wall relaxation with respect to time. The relaxation time can further help in the diagnosis of disease. The pulse wave velocity component is derived by transforming governing fluid flow equations into hyperbolic equations for laminar incompressible blood flow in an artery of viscoelastic walls. The viscoelastic behavior of the wall is analogously modeled by the modified Zener model that has the capability to measure creep, stress relaxation, and hysteresis. The derived model equations are solved numerically by the multiderivative Runge–Kutta implicit–explicit time integration method with a weighted, essentially non-oscillatory, discretization scheme. The results are well validated with clinical trial-based Riemann problems for the case of elastic walls. It is observed that the proposed modified Zener model is well suitable to identify the location of arterial stiffness where the pulse wave manifests various types of phenomena like discontinuity and reflection.

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

A TIME-INTEGRATED SAMPLER FOR RADIOCARBON ANALYSIS OF AQUATIC METHANE

ABSTRACTFreshwater ecosystems are responsible for a large proportion of global methane emissions to the atmosphere. The radiocarbon (14C) content of this aquatic methane is useful for determining the age and source of this important greenhouse gas. Several methods already exist for the collection of aquatic methane for radiocarbon analysis, but they tend to only sample over short periods of time, which can make them unsuitable for characterizing aquatic methane over longer timespans, and vulnerable to missing short-term events. Here, we describe a new time-integrated method for the collection of aquatic methane that provides samples suitable for radiocarbon analysis, that are representative for periods of up to at least 16 days. We report the results of a suite of tests undertaken to verify the reliability of the method, and the 14C age of aquatic methane from field trials undertaken at sites within Scotland, UK. We believe that this new method provides researchers with a simple approach that is easily deployable and can be used to collect representative time-integrated samples of methane for radiocarbon analysis from a wide range of aquatic environments.

Forced vibrations of a cantilever beam using radial point interpolation methods: A comparison study

Meshless methods are a type of numerical method used to simulate continuum mechanics problems. These methods have been applied to several types of problems, there are a few works using meshless method focused on dynamic problems, but most works study static loading conditions. The current work aims at using two different meshless methods, the Radial Point Interpolation Method (RPIM) and the Natural Neighbours RPIM (NNRPIM), in a dynamic problem, specifically forced vibrations. This problem requires time integration, therefore three different time integration methods have also been tested, namely: the Central Difference Method (CDM), the Wilson method, and the Newmark method. The CDM is an explicit method, while the other two are implicit. A discretization study was performed to assess the ideal nodal discretization before the numerical and time integration methods are validated. For the implicit methods, different time step lengths were also tested. In the final example damping was introduced. The results prove the validity of the two meshless methods by having similar results to the Finite Element Method (FEM) using the three distinct time integration methods, different loading conditions, and damping.

Implicit-explicit time integration for the immersed wave equation

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation.We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.

Hybrid methods for solving structural geometric nonlinear dynamic equations: Implementation of fifth-order iterative procedures within composite time integration methods

This paper studies a class of hybrid methods that implement multi-point iterative procedures as the nonlinear solver within optimized composite implicit methods. These multi-point methods are employed to enhance convergence order and accuracy without requiring higher-order derivatives for solving structural geometric nonlinear dynamic equations. The promising approach provides an opportunity to delve deeper into its implications and explore the potential applications of multi-point techniques and composite methods across the analysis of dynamical systems. This study involves the utilization of two-point and five-point iterative methods with fifth-order convergence within three-step and four-step optimized composite implicit time integration methods. Moreover, a comprehensive hybrid scheme is introduced by integrating a family of optimized composite implicit methods with single-point and multi-point iterative methods. Then, the performance of the hybrid scheme is analyzed through structural examples, considering factors such as time step size, tolerance threshold, and the degrees of freedom of structures. The findings and numerical results illustrate that multi-point methods outperform the single-point method in terms of the number of iterations. Simultaneously, the four-step time integration method exhibits superior performance compared to the three-step time integration method, albeit with a marginal difference. It is observed that the hybrid method, consisting of the four-step time integration method and the five-point iterative method, performs better than other methods in terms of the number of iterations and errors for solving geometric nonlinear dynamic equations, especially in large structures. Thus, it stands out as a favorable choice for practical analyses and a possible candidate for generalization to other types of nonlinear problems in computational mechanics.

A Memory-Efficient Neural Ordinary Differential Equation Framework Based on High-Level Adjoint Differentiation

Neural ordinary differential equations (neural ODEs) have emerged as a novel network architecture that bridges dynamical systems and deep learning. However, the gradient obtained with the continuous adjoint method in the vanilla neural ODE is not reverse-accurate. Other approaches suffer either from an excessive memory requirement due to deep computational graphs or from limited choices for the time integration scheme, hampering their application to large-scale complex dynamical systems. To achieve accurate gradients without compromising memory efficiency and flexibility, we present a new neural ODE framework, PNODE, based on high-level discrete adjoint algorithmic differentiation. By leveraging discrete adjoint time integrators and advanced checkpointing strategies tailored for these integrators, PNODE can provide a balance between memory and computational costs, while computing the gradients consistently and accurately. We provide an open-source implementation based on PyTorch and PETSc, one of the most commonly used portable, scalable scientific computing libraries. We demonstrate the performance through extensive numerical experiments on image classification and continuous normalizing flow problems. We show that PNODE achieves the highest memory efficiency when compared with other reverse-accurate methods. On the image classification problems, PNODE is up to two times faster than the vanilla neural ODE and up to 2.3 times faster than the best existing reverse-accurate method. We also show that PNODE enables the use of the implicit time integration methods that are needed for stiff dynamical systems.

Proper Orthogonal Decomposition Based Intrusive Reduced Order Models to Accelerate Computational Speed of Dynamic Analyses of Structures Using Explicit Time Integration Methods

Proper Orthogonal Decomposition Based Intrusive Reduced Order Models to Accelerate Computational Speed of Dynamic Analyses of Structures Using Explicit Time Integration Methods

Microwave time reversal for nondestructive testing of buried small damage in composite materials

Composite materials are widely applied in aerospace, civil engineering, and sports equipment. Various damages produced during fabrication and long-term use can destroy its original mechanical properties, which brings safety and structural healthy concerns. Microwave imaging based on time reversal (TR) is one of the most promising nondestructive testing methods for portable, low-cost, and accurate testing with the advantages of auto-focus and super-resolution. This paper applied microwave TR for the detection of buried small damage in composites backed by metal plates. Strong reflection from composite–metal interfaces brings challenges in successfully achieving time-reversal auto-focusing on small and weak-scattering damages in composites. Traditional target localization methods, including the entropy regularization method and time-integrated energy method, may result in the wrong localization of small damages. The main contribution of this paper is that the localization problem caused by the strong reflection from metal plates is revealed first, and the target initial reflection method from through-wall-radar imaging is introduced to solve it. The performance of three target localization methods is investigated, and the physical reasons for failure or successful localization are discussed in detail. Some performance influence factors, such as the arrangement of receivers or the total time step of received signals, are also discussed. Good performance for the detection of a single small damage with a weak scattered signal is achieved, and the performance for detecting multiple damages is studied. All time-reversal simulations are carried out based on the finite-difference time-domain method.

A modified Chebyshev collocation method for the generalized probability density evolution equation

Solving the generalized density evolution equation (GDEE) is essential to obtain the structural random response by the probability density evolution method (PDEM). To improve the collocation method, this study proposes a modified Chebyshev collocation method (MCCM) scheme for the solution of the GDEE based on the pseudo-spectral method, and derives a refine time integration format for GDEE. The GDEE is numerically solved by combining the fine time integration method and MCCM, and the accuracy, efficiency and numerical stability of the algorithm are investigated. Two numerical examples indicate that the results for the proposed method are in good agreement with analytical solutions and Monte Carlo simulation results. The proposed four numerical collocation schemes could effectively suppress numerical dispersion and numerical dissipation. In addition, the modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency.

An efficient fully Crouzeix-Raviart finite element model for coupled hydro-mechanical processes in variably saturated porous media

Coupled hydro-mechanical processes in variably saturated porous media can be encountered in several applications in geosciences and soil mechanics. They are ruled by highly nonlinear and strongly coupled equations of fluid flow in unsaturated porous media and quasi-static mechanical deformation. In this work, we develop an efficient and robust numerical scheme for unsaturated poroelasticity that allows for balancing computational requirements and accuracy. The spatial discretization is based on the Crouzeix-Raviart (CR) finite element method for both the displacement field and the hydraulic head. The CR method is locally mass conservative and uses low-order approximation, which alleviates the computational burden. The time discretization of the coupled nonlinear hydraulic mechanical equations is performed using high order time integration methods and efficient time stepping schemes via the method of lines (MOL).Four test cases are investigated to highlight the efficiency and accuracy of the developed numerical scheme. The first test case deals with the settlement induced by gravity in the case of saturated and unsaturated soils. A new semi-analytical solution is developed for the settlement in the unsaturated case and used for the validation of the numerical scheme. The second test case is the Cantilever bracket problem, extended here to variably saturated media and studied to investigate the nonphysical oscillations of the pressure variable with different numerical schemes. The results are compared to finite element solutions obtained with COMSOL with either linear or quadratic approximation of the displacement field. The third test case deals with the deformation of an unsaturated soil under partial infiltration and surface load. This test case is simulated to investigate the accuracy and efficiency of the CR scheme for the challenging problem of infiltration into dry soils. The last test case is simulated to show the applicability of the developed model to a realistic problem dealing with the stability of a hillslope under intense rainfall periods.Results of the numerical experiments show that the CR scheme coupled with high order time integration methods allows for significant gain in both computational requirements and accuracy.

Numerical coupling of aerosol emissions, dry removal, and turbulent mixing in the E3SM Atmosphere Model version 1 (EAMv1) – Part 2: A semi-discrete error analysis framework for assessing coupling schemes

Abstract. Part 1 (Wan et al., 2024) of this study discusses the motivation and empirical evaluation of a revision to the aerosol-related numerical process coupling in the atmosphere component of the Energy Exascale Earth System Model version 1 (EAMv1) to address the previously reported issue of strong sensitivity of the simulated dust aerosol lifetime and dry removal rate to the model's vertical resolution. This paper complements that empirical justification of the revised scheme with a mathematical justification leveraging a semi-discrete analysis framework for assessing the splitting error of process coupling methods. The framework distinguishes the error due to numerical splitting from the error due to the time integration method(s) used for each individual process. Such a distinction results in a framework that provides an intuitive understanding of the causes of the splitting error. The application of this framework to the dust life cycle in EAMv1 confirms (i) that the original EAMv1 scheme artificially strengthens the effect of dry removal processes and (ii) that the revised splitting reduces that artificial strengthening. While the error analysis framework is presented in the context of the dust life cycle in EAMv1, the framework can be broadly leveraged to evaluate process coupling schemes, both in other physical problems and for any number of processes. This framework will be particularly powerful when the various process implementations support a variety of time integration approaches. Whereas traditional local truncation error approaches require separate consideration of each combination of time integration methods, this framework enables evaluation of coupling schemes independent of particular time integration approaches for each process while still allowing for the incorporation of these specific time integration errors if so desired. The framework also explains how the splitting error terms result from (i) the integration of individual processes in isolation from other processes and (ii) the choices of input state and time step size for the isolated integration of processes. Such a perspective has the potential for the rapid development of alternative coupling approaches that utilize knowledge both about the desired accuracy and about the computational costs of individual processes.