Time Integration Scheme Research Articles

Flow dynamics over cylinder in two‐dimensional benchmark problem: A finite element approximation

This research investigates 2D benchmark flow around a circular cylinder, utilizing the incompressible Navier–Stokes equations alongside the continuity and energy equations. The numerical solution is achieved through finite element discretization for space variable, combined with a second‐order Crank–Nicolson scheme for time integration. The computational results are derived using the FEATFLOW finite element based library package. Our study focuses on the dimensionless form of the flow equations and examines three key dimensionless parameters: drag, lift, and pressure drop. Upon applying finite element method (FEM) discretization, the system is converted into a set of linear or nonlinear ordinary differential equations or algebraic equations for steady‐state scenarios. We then apply the Newton–Raphson method as the outer nonlinear solver, and the multigrid method for efficiently resolving the linear subproblems. To ensure numerical accuracy, we evaluated the and errors, confirming that the experimental order of convergence matches the theoretical predictions. Flow profiles were both graphically represented and tabulated, offering a detailed understanding of the simulation results.

An open-source time-domain wave-based room acoustic software in Python based on the nodal discontinuous Galerkin method

In this work, we introduce an open-source implementation of a time-domain wave-based room acoustic modeling software package, named DG_RoomAcoustics. In this software, the linear acoustic equations are spatially discretized by the nodal discontinuous Galerkin method, and are integrated in time by either the explicit Runge-Kutta or the arbitrary high-order derivatives (ADER) integration schemes. Following the principles of object-oriented programming paradigm, the software is structured to ensure generic applicability and to facilitate future extensions with additional functionalities (e.g., different time integration schemes, boundary conditions). A comprehensive presentation of the physical and numerical aspects of the problem is provided. A detailed exposition of the code structure and components is presented, all of which are released under an open-source license, fostering community feedback and collaborative contributions for ongoing improvements. A brief overview of the current capabilities of the software is introduced. Future work regarding possible functionality extensions and performance optimizing are discussed.

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.

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.

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.

Second-order mean-field approximation for calculating dynamics in Au-nanoparticle networks.

Exploiting physical processes for fast and energy-efficient computation bears great potential in the advancement of modern hardware components. This paper explores nonlinear charge tunneling in nanoparticle networks, controlled by external voltages. The dynamics are described by a master equation, which expresses the time-evolution of a distribution function over the set of charge occupation numbers. The driving force behind this evolution is charge tunneling events among nanoparticles and their associated rates. We introduce two mean-field approximations to this master equation. By parametrization of the distribution function using its first- and second-order statistical moments, and a subsequent projection of the dynamics onto the resulting moment manifold, one can deterministically calculate expected charges and currents. Unlike a kinetic Monte Carlo approach, which extracts samples from the distribution function, this mean-field approach avoids any random elements. A comparison of results between the mean-field approximation and an already available kinetic Monte Carlo simulation demonstrates great accuracy. Our analysis also reveals that transitioning from a first-order to a second-order approximation significantly enhances the accuracy. Furthermore, we demonstrate the applicability of our approach to time-dependent simulations, using Eulerian time-integration schemes.

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.

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.

Numerical model for time-dependent cracking and deflection of large-span concrete bridge structures

A new computational framework is proposed to analyse the simultaneous occurrence of multiple physical processes affecting the long-term behavior of large-scale concrete structures. Relying on the additivity of small strain, the framework comprehensively takes the nonlinearity of concrete creep, cyclic creep induced by traffic load, concrete shrinkage, concrete cracking, normal-strength steel rebars and prestressing strands relaxation into account, and other physical processes are further imagined. The algorithm is implemented with Abaqus/Standard with the aid of several user-supplied subroutines. The implicit time integration scheme is deemed appropriate for large-scale structures. The framework is systematically validated by several classical experimental results. Finally, a three-dimensional study of a three-span continuous rigid frame bridge with 270 m main span is performed. Comparison between the experimental and numerical results further reveals the contributions of various potential influencing factors. The proposed model shows promise for predicting the time-dependent performance of large-scale reinforcement concrete (RC) structures.

Nonlinear dynamic response of a sandwich plate with negative Poisson’s ratio honeycomb-core layer under low-velocity collision impact

In this paper, a nonlinear analytical model is presented for the low-velocity collision impact of a sandwich plate with auxetic honeycomb-core layer. The auxetic feature of the honeycomb material is realized by mathematically expressing the effective material coefficients in terms of material property and cellular geometric parameters through a homogenization method. The higher order shear deformation theory, the von Kármán nonlinearity theory, and a modified Hertz contact law which accounts for the contact pressure distribution and indentation effect, are employed to establish the kinematic relations. The Newmark time integration scheme in conjunction with the direct iterative method is utilized to establish a solution procedure for the nonlinear dynamic governing equation. The verification of the presented model with the data in published literatures is carried out, followed by a series of numerical analyses for influences of cellular geometric features and impactor’s initial conditions (such as impactor’s initial velocity, mass, and nose curvature radius) on the nonlinear dynamic response. The results of numerical analyses show that the geometric features of the unit cell in the honeycomb-core and impactor’s initial conditions can cause significant influences on the nonlinear collision impact behaviors of the system.

Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations

We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen–Cahn and the Fisher–KPP equations on the manifold of fixed-rank matrices. Our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

Implicit-explicit Runge-Kutta for radiation hydrodynamics I: Gray diffusion

Radiation hydrodynamics are a challenging multiscale and multiphysics set of equations. To capture the relevant physics of interest, one typically must time step on the hydrodynamics timescale, making explicit integration the obvious choice. On the other hand, the coupled radiation equations have a scaling such that implicit integration is effectively necessary in non-relativistic regimes. A first-order Lie-Trotter-like operator split is the most common time integration scheme used in practice, alternating between an explicit hydrodynamics step and an implicit radiation solve and energy deposition step. However, such a scheme is limited to first-order accuracy, and nonlinear coupling between the radiation and hydrodynamics equations makes a more general additive partitioning of the equations non-trivial. Here, we develop a new formulation and partitioning of radiation hydrodynamics with gray diffusion that allows us to apply (linearly) implicit-explicit Runge-Kutta time integration schemes. We prove conservation of total energy in the new framework, and demonstrate 2nd-order convergence in time on multiple radiative shock problems, achieving error 3–5 orders of magnitude smaller than the first-order Lie-Trotter operator split at the hydrodynamic CFL, even when Lie-Trotter applies a 3rd-order TVD Runge-Kutta scheme to the hydrodynamics equations.

Adaptive conservative time integration for unsteady compressible flow

Unsteady flow computations typically rely on time integration schemes which employ the same step size everywhere. However, in cases with strong variations of wave speed and/or spatial resolution, local stability criteria and truncation errors would allow for much larger time steps in parts of the domain. To exploit this potential of improving the computational efficiency, adaptive time-stepping methods have been developed. Recently, an adaptive conservative time integration (ACTI) scheme for explicit finite volume methods was devised. Opposed to previous methods, ACTI guarantees exact conservation and periodic synchronization. Both are achieved by splitting a global time step into 2L (L≥0 is a cell dependent level) sub-steps, whereas the order of cell updates is critical. It has been demonstrated for flows in fractured porous media and for the Euler equations that ACTI can reduce the computational cost dramatically while preserving second order accuracy in space and time. In this paper, the ACTI scheme is extended to the compressible Navier-Stokes-Fourier system, and special attention is required for the spatio-temporal discretization of the convective and viscous fluxes. Numerical studies of unsteady flows involving shock boundary layer interaction demonstrate that the same accuracy is achieved with the new compressible ACTI flow solver as with classical time integration, but at much lower cost. Moreover, since the method is explicit, it has the potential for efficient computations on multiple CPUs and GPUs.

Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations

In this paper we construct a novel discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations. The Cahn-Hilliard equation models the separation of a binary mixture. We construct a very simple time integration scheme for simulating the Cahn-Hilliard equation, which is based on splitting the fourth-order equation into two second-order Helmholtz equations. We combine the Cahn-Hilliard equation with the Navier-Stokes equations to simulate phase separation in a two-phase fluid flow in two dimensions. The scheme conserves mass and momentum and exhibits consistency between mass and momentum, allowing it to be used with large density ratios. We introduce a novel discretization of the surface tension force from the phase-field variable that has finite support around the transition region. The model has a parameter that allows it to transition from a smoothed continuum surface force to a fully sharp interface formulation. We show that our method achieves second-order accuracy, and we compare our method to previous work in a variety of experiments.

Multi-step Hermite-Birkhoff predictor-corrector schemes

In this study, we introduce a multi-step multi-derivative predictor-corrector time integration scheme analogous to the schemes in Schütz et al. (2022) [13], incorporating a multi-step quadrature rule. We conduct stability analysis up to order eight and optimize the schemes to achieve A(α)-stability for large α. Numerical experiments are performed on ordinary differential equations exhibiting diverse stiffness conditions, as well as on partial differential equations showcasing non-linearity and higher-order terms. Results demonstrate the convergence and flexibility of the proposed schemes across diverse situations.

Rocking block simulation based on numerical dissipation.

In this paper, a computational approach based on numerical dissipation is proposed to simulate rocking blocks. A rocking block is idealized as a solid body interacting with its foundation through a contact-based formulation. An implicit time integration scheme with numerical dissipation, set to optimally treat dissipation in contact problems, is employed. The numerical dissipation is ruled by the time step and the rocking dissipative phenomenon at impacts is accurately predicted without any damping model. A broad numerical campaign is conducted to define a regression law in analytic form for the setting of the time step, depending on the block size and aspect ratio, the contact stiffness, as well as the coefficient of restitution selected. The so-obtained regression law appears accurate and an a posteriori validation with cases not in the training dataset confirms the effectiveness of the approach. Finally, the comparison with available experimental tests highlights the approach efficacy for free rocking and harmonic loading cases (in a deterministic sense), and for earthquake-like loading cases (in a statistical sense). It is found that rocking blocks with sizes of interest for structural engineering (e.g., cultural heritage structures) can be simulated with time steps within 10-3 ÷ 10-1s, so allowing very fast computations.

Projective integration methods in the Runge–Kutta framework and the extension to adaptivity in time

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge–Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge–Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge–Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge–Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

Simplicits: Mesh-Free, Geometry-Agnostic Elastic Simulation

The proliferation of 3D representations, from explicit meshes to implicit neural fields and more, motivates the need for simulators agnostic to representation. We present a data-, mesh-, and grid-free solution for elastic simulation for any object in any geometric representation undergoing large, nonlinear deformations. We note that every standard geometric representation can be reduced to an occupancy function queried at any point in space, and we define a simulator atop this common interface. For each object, we fit a small implicit neural network encoding spatially varying weights that act as a reduced deformation basis. These weights are trained to learn physically significant motions in the object via random perturbations. Our loss ensures we find a weight-space basis that best minimizes deformation energy by stochastically evaluating elastic energies through Monte Carlo sampling of the deformation volume. At runtime, we simulate in the reduced basis and sample the deformations back to the original domain. Our experiments demonstrate the versatility, accuracy, and speed of this approach on data including signed distance functions, point clouds, neural primitives, tomography scans, radiance fields, Gaussian splats, surface meshes, and volume meshes, as well as showing a variety of material energies, contact models, and time integration schemes.

A performance study of horizontally explicit vertically implicit (HEVI) time-integrators for non-hydrostatic atmospheric models

We conduct a thorough study of different forms of horizontally explicit and vertically implicit (HEVI) time-integration strategies for the compressible Euler equations on spherical domains typical of nonhydrostatic global atmospheric applications. We compare the computational time and complexity of two nonlinear variants (NHEVI-GMRES and NHEVI-LU) and a linear variant (LHEVI). We report on the performance of these three variants for a number of additive Runge-Kutta methods ranging in order of accuracy from second through fifth, and confirm the expected order of accuracy of the HEVI methods for each time-integrator. To gauge the maximum usable time-step of each HEVI method, we run simulations of a nonhydrostatic baroclinic instability for 100 days and then use this time-step to compare the time-to-solution of each method. The results show that NHEVI-LU is 3x faster than NHEVI-GMRES, and LHEVI is 8x faster than NHEVI-GMRES, for the idealized cases tested. The baroclinic instability and inertia-gravity wave simulations indicate that the optimal choice of HEVI method is LHEVI. For the fastest time-to-solution, the second and third order methods are best although the better accuracy of the high-order methods (particularly the fourth order method) should be considered.In the future, we will report on whether these results hold for more complex problems using, e.g., real atmospheric data and/or a higher model top typical of space weather applications.

Simulation of impacts between spherical rigid bodies with frictional effects

AbstractThis work studies the impact between spherical rigid bodies in the frame of nonsmooth contact dynamics considering friction effects. A new impact element formulation based on the classical instantaneous local Newton impact law is presented. The kinematics properties of the spheres are described by a rigid body formulation with translational and rotational degrees of freedom referred to an inertial frame. In addition, an extension of the nonsmooth generalized‐ time integration scheme applied to collisions with multiple impacts including Coulomb's friction law is given. Six numerical examples are presented to evaluate the robustness and the performance of the proposed methodology.