Time Integration Research Articles

Swirling electrolyte. Part 2. Secondary circulation and its stability

The asymptotic analysis of steady azimuthally invariant electromagnetically driven flows occurring in a shallow annular layer of electrolyte undertaken in Part 1 of this study (McCloughan & Suslov, J. Fluid Mech., vol. 980, 2024, A59) predicted the existence of a two-tori flow state that has not been detected previously. In Part 2 of the study we confirm its existence by numerical time integration of the governing equations. We observe a hysteresis, where the type of solution obtained for the same set of governing parameters depends on the choice of the initial conditions and the way the governing parameters change, which is fully consistent with the analytic results of Part 1. Subsequently, we perform a linear stability analysis of the newly obtained steady state and deduce that the experimentally observed anti-cyclonic free-surface vortices appear on its background as a result of a centrifugal (Rayleigh-type) instability of the interface separating two counter-rotating toroidal structures that form the newly found flow solution. The quantitative characteristics of such instability structures are determined. It is shown that such structures can only exist in sufficiently thin layers with the depth not exceeding a certain critical value.

Optimisation of the hammer throw using parameterised synthetic motion kinematics in a multi‐body simulation

AbstractHammer throwing is a motorically demanding sport in which the execution of the movement significantly influences the result, that is the throwing distance. There are many approaches for supporting technical training that are based on measurement technology and image capturing. The accelerations and locus curves of the athlete and hammer are recorded and compared. In this article, these methods are supplemented by a multi‐body simulation (MBS) of the hammer throw. The kinematics of the hand movement is described using a parameterized synthetic trajectory. Scalar parameters in the approach allow the trajectory to be modified. With a MBS model of the wire cable and the hammer, the movement of the hammer can be determined by numerical time integration from the hand's trajectory. As a result the final throwing distance after free flight phase can be analyzed. Due to the complex spatial motion of the hammer during the acceleration phase only heuristic methods can be used to optimize the maximum throwing distance. In a final step the optimized movement can be used to improve the athlete's training.

On the adaptive solution of phase‐field problems with A‐stable explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods

AbstractRunge–Kutta algorithms with adaptive step size control provide reliable tools for the solution of initial value problems with diagonally implicit Runge‐Kutta (DIRK) methods as the most common approach. In this contribution, the new low‐order explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods are investigated, combining implicit schemes with an additional explicit evaluation as an explicit last stage. This results in Butcher tableaus with two solutions of different convergence orders suitable for embedded methods, where the higher‐order solution is achieved by additional explicit evaluations. Thus, the iterative solution of non‐linear systems is omitted for the additional stage, presenting a major reduction in computational cost for the determination of a local error estimate. The key contribution is the application of the novel Butcher tableaux to phase‐field problems, solved with the finite‐element method, leading to substantial numerical investigations with an efficient approach for DIRK schemes. The most important aspects are the investigations of stability properties which lead to the novel class of A‐stable ELDIRK methods. In accordance, the study of the convergence orders is presented. A local error estimator is presented, such that adaptive step size control for the new low‐order embedded schemes based on an empirical approach for error estimation is achieved. A suitable parallel algorithm is presented with conclusive two‐dimensional phase‐field simulations based on a Kobayashi‐Warren‐Carter model. The higher‐order convergence suggested by the novel schemes is confirmed, and their effective results are demonstrated, resulting in a valuable semi‐explicit addition to the family of Runge–Kutta time integration schemes.

On the measurement of the nonlinear dynamics of sandwich sector plate surrounded by the auxetic concrete foundation: Introducing a machine learning algorithm for nonlinear problems

The measurement of the nonlinear dynamics of sandwich rotary sector plates is crucial for measurement engineers as it enables the precise analysis and understanding of the behavior of advanced composite structures under dynamic conditions. These measurements help in identifying and mitigating potential issues related to vibration, stability, and fatigue, which are critical in industries like aerospace, automotive, and civil engineering. Accurate data on nonlinear dynamics aids in the optimization of design, enhancing performance, durability, and safety. Furthermore, it supports the development of predictive maintenance strategies, reducing downtime and operational costs, and contributing to the advancement of engineering materials and techniques. So, in the current work, for the first time, nonlinear dynamics of sandwich rotary sector plate via a mathematical modeling and machine learning algorithm is presented. First, due to a lack of information on the nonlinear dynamics of sandwich rotary sector plates, a dataset for training, testing, and validating the machine learning algorithm is presented. For this purpose, using Hamilton’s principle, Von-Karman nonlinearity, and first-order shear deformation theory (FSDT), the nonlinear governing equations are obtained. Also, due to increasing stiffness and stability, an auxetic concrete foundation covers the sandwich structure. After that using the two-dimensional generalized differential quadrature method (GDQM) and Newmark’s time integration method (NTIM), the nonlinear equations are solved. This research provides valuable insights for engineers in designing advanced composite structures with improved dynamic properties. The results also contribute to the broader understanding of nonlinear dynamic interactions in complex material systems, paving the way for innovative applications in various engineering fields.

Comparison of different elastic strain definitions for largely deformed SEI of chemo-mechanically coupled silicon battery particles

Amorphous silicon is a highly promising anode material for next-generation lithium-ion batteries. Large volume changes of the silicon particle have a critical effect on the surrounding solid-electrolyte interphase (SEI) due to repeated fracture and healing during cycling. Based on a thermodynamically consistent chemo-elasto-plastic continuum model we investigate the stress development inside the particle and the SEI. Using the example of a particle with SEI, we apply a higher order finite element method together with a variable-step, variable-order time integration scheme on a nonlinear system of partial differential equations. Starting from a single silicon particle setting, the surrounding SEI is added in a first step with the typically used elastic Green–St-Venant (GSV) strain definition for a purely elastic deformation. For this type of deformation, the definition of the elastic strain is crucial to get reasonable simulation results. In case of the elastic GSV strain, the simulation aborts. We overcome the simulation failure by using the definition of the logarithmic Hencky strain. However, the particle remains unaffected by the elastic strain definitions in the particle domain. Compared to GSV, plastic deformation with the Hencky strain is straightforward to take into account. For the plastic SEI deformation, a rate-independent and a rate-dependent plastic deformation are newly introduced and numerically compared for three half cycles for the example of a radial symmetric particle.

Low regularity error estimates for the time integration of 2D NLS

Low regularity error estimates for the time integration of 2D NLS

Effective highly accurate time integrators for linear Klein–Gordon equations across the scales

Effective highly accurate time integrators for linear Klein–Gordon equations across the scales

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.

Peridynamics modelling of projectile penetration into concrete targets

A non-ordinary state-based peridynamics (NOSB-PD) model is proposed to simulate the projectile penetration into concrete targets. In this model, the Kong-Fang concrete material model recently proposed is firstly implemented into the NOSB-PD framework to describe the complex dynamic behavior and failures in concrete material subjected to penetration loading, and then an improved point-to-volume discrete frictional contact model is proposed to simulate the physical interaction between projectile and target. After the mesh-free discretization and explicit time integration, the proposed NOSB-PD model is used to numerically predict two sets of projectile penetration experiments into low-strength and high-strength concrete targets. And numerical predictions are found to be in good agreements with corresponding test data including penetration depth, projectile deceleration, deformation of projectile and failures in concrete targets.

Stochastic Dynamic Buckling Analysis of Cylindrical Shell Structures Based on Isogeometric Analysis

In this paper, we extend our previous work on the dynamic buckling analysis of isogeometric shell structures to the stochastic situation where an isogeometric deterministic dynamic buckling analysis method is combined with spectral-based stochastic modeling of geometric imperfections. To be specific, a modified Generalized-α time integration scheme combined with a nonlinear isogeometric Kirchhoff–Love shell element is used to simulate the buckling and post-buckling problems of cylindrical shell structures. Additionally, geometric imperfections are constructed based on NURBS surface fitting, which can be naturally incorporated into the isogeometric analysis framework due to its seamless CAD/CAE integration feature. For stochastic analysis, the method of separation is adopted to model the stochastic geometric imperfections of cylindrical shells based on a set of measurements. We tested the accuracy and convergence properties of the proposed method with a cylindrical shell example, where measured geometric imperfections were incorporated. The ABAQUS reference solutions are also presented to demonstrate the superiority of the inherited smooth and high-order continuous properties of the isogeometric approach. For stochastic dynamic buckling analysis, we evaluated the buckling load variability and reliability functions of the cylindrical shell with 500 samples generated based on seven nominally identical shells reported in the geometric imperfection data bank. It is noted that the buckling load variability in the cylindrical shell obtained with static nonlinear analysis is also presented to show the differences between dynamic and static buckling analysis.

Numerical challenges for energy conservation inN-body simulations of collapsing self-interacting dark matter halos

Context.Dark matter (DM) halos can be subject to gravothermal collapse if the DM is not collisionless, but engaged in strong self-interactions instead. When the scattering is able to efficiently transfer heat from the centre to the outskirts, the central region of the halo collapses and reaches densities much higher than those for collisionless DM. This phenomenon is potentially observable in studies of strong lensing. Current theoretical efforts are motivated by observations of surprisingly dense substructures. However, a comparison with observations requires accurate predictions. One method to obtain such predictions is to useN-body simulations. Collapsed halos are extreme systems that pose severe challenges when applying state-of-the-art codes to model self-interacting dark matter (SIDM).Aims.In this work, we investigate the root of such problems, with a focus on energy non-conservation. Moreover, we discuss possible strategies to avoid them.Methods.We ranN-body simulations, both with and without SIDM, of an isolated DM-only halo and we adjusted the numerical parameters to check the accuracy of the simulation.Results.We find that not only the numerical scheme for SIDM can lead to energy non-conservation, but also the modelling of gravitational interaction and the time integration are problematic. The main issues we find are: (a) particles changing their time step in a non-time-reversible manner; (b) the asymmetry in the tree-based gravitational force evaluation; and (c) SIDM velocity kicks breaking the time symmetry.Conclusions.Tuning the parameters of the simulation to achieve a high level of accuracy allows us to conserve energy not only at early stages of the evolution, but also later on. However, the cost of the simulations becomes prohibitively large as a result. Some of the problems that make the simulations of the gravothermal collapse phase inaccurate can be overcome by choosing appropriate numerical schemes. However, other issues still pose a challenge. Our findings motivate further works on addressing the challenges in simulating strong DM self-interactions.

An improved Arbitrary Lagrangian–Eulerian thermal-fluid model by considering powder deposition effects on melting pool during Direct Energy Deposition processes

Directed Energy Deposition (DED) has emerged notably by offering new possibilities for (re-)manufacturing parts. The multi-phase thermal-fluid models that simulate powder stream deposition are computationally too expensive. The mono-phase Arbitrary Lagrangian–Eulerian (ALE) method are limited in prediction due to exclusion of powder deposition parameters. To address this, we propose an improved 3D mono-phase thermal-fluid model that incorporates powder deposition effects and employs a Moving Thermal-Fluid (MTF) framework to accelerate simulations. Mass, energy, and momentum conservation equations are solved using the finite element method (FEM) and an implicit time integration algorithm, and the ALE method tracks the free surface during deposition. The improved ALE allows considering enthalpy and momentum related to powder deposition through the implementation of new source terms in the energy and momentum conservation equations, leading to more accurate predictions without significantly increasing computing time. Numerical investigations into powder deposition parameters, such as powder distribution, powder enthalpy, and powder momentum, are conducted to identify their impact on melting pool prediction. The in-situ and ex-situ measurements of the melting pool are also performed to check the efficiency of the proposed model. The applications highlight the importance of incorporating powder stream effects and demonstrate the proposed model’s computational efficiency compared to the classical ALE model.

Coupling a time-domain code with a RANS solver for slamming analysis

ABSTRACT A time domain hydroelastic code (TD) based on strip theory is two-way coupled with a Reynolds Averaged Navier−Stokes (RANS) solver to capture the bottom slamming effects on ships. The study is conducted on an Ultra Large Container Ship (ULCS). The methodology considers the hull’s flexibility and analyses longitudinal bending in head seas. Ship motions are calculated using a body non-linear time-domain method based on strip theory. The equations of motion are solved, and time integration is performed using the fourth-order Runge–Kutta method. When a slamming event is identified, the TD code activates a RANS solver to determine the bottom slamming force. The simulations are conducted under extreme sea conditions, and the hull experiences springing and whipping responses. The numerical results of vertical bending moments are compared with experimental data. The bottom and bow flare slamming are identified from the simulations.

Forced vibration characteristics of viscoelastic variable stiffness laminated composite plates using time and frequency domain approaches

The linear forced vibration characteristics of viscoelastic variable stiffness laminated composite plates (VVSLC) are studied using a generalized Maxwell model and finite element method. The integral form of the viscoelastic constitutive relation is converted to the incremental form for finite element formulation based on the Reissner–Mindlin plate theory. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response directly in the time domain is obtained using shooting technique coupled with Newmark’s time integration method. The implementation of shooting technique for Boltzmann integral-based viscoelasticity for curvilinear fibre composite plates is done for the first time in this study. For the comparison purpose, the response/resonance frequency/modal loss factor is also obtained using equivalent complex modulus based viscoelastic correspondence principle. It is observed that the variation in fibre orientation and boundary conditions leads to significant variations in response, stress/moment resultant amplitude and the damping factor of the VVSLC plates. Further, the present time domain based approach is capable of predicting damping factor at all forcing frequencies whereas complex eigenvalue analysis can predict damping factor only at discrete resonance frequency. Based on the detailed studies, it is found that the curvilinear fibre composite plates depict a significant reduction in response/moment resultants compared to straight fibre composite plates.

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios χ∥/χ⊥⋙1 (with χ∥, χ⊥ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. The implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ∼(NΔt)1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios (χ∥/χ⊥=1010).

SMS: Spiking marching scheme for efficient long time integration of differential equations

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.

A dual-modified implicit time integration method for three-dimensional impact modelling within the framework of the SBFEM

This paper presents a dual-modified implicit time integration method (DMITIM) for three-dimensional impact modelling within the framework of the scaled boundary finite element method (SBFEM). Controlled by a single parameter, this method effectively dissipates high-frequency responses in impact problems without affecting low-frequency responses, ensuring time integration stability. To address the issue of false oscillations during impacts, additional Lagrange multipliers are introduced to supplement the velocity and acceleration corrections at each time integration step. For nonmatching meshes, by leveraging the mesh flexibility of the SBFEM, a rapid and straightforward remeshing method is utilized to establish a node-to-node contact scheme. The numerical results demonstrate that the DMITIM significantly reduces false oscillations in the velocity and contact force during impact events. It also showcases the flexibility of the SBFEM in handling nonmatching meshes. The combination of the SBFEM with the DMITIM provides enhanced simulation accuracy and stability for impact problems compared to the results from ABAQUS using the HHT-α time integration algorithm.

Domain adaptation strategies for 3D reconstruction of the lumbar spine using real fluoroscopy data

In this study, we address critical barriers hindering the widespread adoption of surgical navigation in orthopedic surgeries due to limitations such as time constraints, cost implications, radiation concerns, and integration within the surgical workflow. Recently, our work X23D showed an approach for generating 3D anatomical models of the spine from only a few intraoperative fluoroscopic images. This approach negates the need for conventional registration-based surgical navigation by creating a direct intraoperative 3D reconstruction of the anatomy. Despite these strides, the practical application of X23D has been limited by a significant domain gap between synthetic training data and real intraoperative images.In response, we devised a novel data collection protocol to assemble a paired dataset consisting of synthetic and real fluoroscopic images captured from identical perspectives. Leveraging this unique dataset, we refined our deep learning model through transfer learning, effectively bridging the domain gap between synthetic and real X-ray data. We introduce an innovative approach combining style transfer with the curated paired dataset. This method transforms real X-ray images into the synthetic domain, enabling the in-silico-trained X23D model to achieve high accuracy in real-world settings.Our results demonstrated that the refined model can rapidly generate accurate 3D reconstructions of the entire lumbar spine from as few as three intraoperative fluoroscopic shots. The enhanced model reached a sufficient accuracy, achieving an 84% F1 score, equating to the benchmark set solely by synthetic data in previous research. Moreover, with an impressive computational time of just 81.1 ms, our approach offers real-time capabilities, vital for successful integration into active surgical procedures.By investigating optimal imaging setups and view angle dependencies, we have further validated the practicality and reliability of our system in a clinical environment. Our research represents a promising advancement in intraoperative 3D reconstruction. This innovation has the potential to enhance intraoperative surgical planning, navigation, and surgical robotics.

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

Timestep-dependent element interpolation functions in the method of matched sections on the example of heat conduction problem

The paper is devoted to further elaboration of the method of matched sections as a new technique within the finite element method. Like FEM it supposes that: a) the complex domain is represented as a mesh of nonintersecting simple elements; b) algebraic relations between the main parameters of an element are established from the governing differential equations; c) all relationships from all elements are assembled into one global matrix. On the other hand, it has two distinct features. The first one is that relations between kinematic and inner force parameters (called Connection equations) are derived from the approximate analytical solution of the governing equations rather than by the application of minimization techniques. The second one consists in that the conjugation between elements is provided between the adjacent sides (sections) rather than in the nodes of the elements. In application to the transient 2D heat conduction, 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, T, and heat flux Q. In practical realization for rectangular finite elements, the method is reduced to the determination of eight unknowns for each element – two unknowns on each side, which are related by the connection equations, and the requirement of the temperature continuity at the center of the element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step becomes the parameter of the element interpolation function within the element, i.e. it determines the behavior of the connection equations. This method was initially proposed by the first author for several 1D problems, and here for the first time, it is applied to 2D problems. The number of tests for rectangular plates exhibits the remarkable properties of the proposed time integration scheme concerning stability, accuracy, and absence of any restrictions as to increasing the time step.