Efficient Time Integration Research Articles

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.

Efficient and Stable Time Integration of Cahn–Hilliard Equations: Explicit, Implicit, and Explicit Iterative Schemes

Efficient and Stable Time Integration of Cahn–Hilliard Equations: Explicit, Implicit, and Explicit Iterative Schemes

A Novel Approach to Linear and Nonlinear Time-History Analysis of Structures: Gauss–Lobatto–Hermite 4-Point (GLH-4P) Method

An efficient time integration scheme has been devised for conducting time-history dynamic analysis of single-degree-of-freedom (SDOF) structural systems. The developed scheme belongs to Runge-Kuta family. The primary innovation of the new formulation lies in the constructive cooperation between the Hermite interpolation and the Gauss–Lobatto integration, linked to address the second-order ordinary differential equation of motion (DEOM). To put the idea into practice, several Hermite interpolators are generated to exploit the information of the internal points of Gauss-Lobatto integrator. The 4-point Gauss–Lobatto formula is employed to numerically integrate acceleration and velocity functions. This novel methodology has been designated the Gauss–Lobatto–Hermite 4-point (GLH-4P) method. GLH-4P furnishes a generally applicable implicit algorithm for conducting both linear and nonlinear analyses of structures subjected to seismic excitation. The advantage of the proposed algorithm over the conventional techniques is its higher accuracy level. Besides, it offers better stability compared with the conventional methods such as the Newmark-β and Wilson-θ methods. Unlike the other methods, GLH-4P can handle high-frequency systems without reducing the step size. Numerical examples reveal the efficacy of the GLH-4P method when juxtaposed against existing methodologies.

Numerical Analysis of an Efficient Time Integration Scheme for Unsteady Conjugate Heat Transfer

Numerical Analysis of an Efficient Time Integration Scheme for Unsteady Conjugate Heat Transfer

On numerical approaches to nonlinear Schrödinger and Korteweg–de Vries equations for piecewise smooth and slowly decaying initial data

We discuss numerical approaches for nonlinear Schrödinger and Korteweg–de Vries equations with slowly decaying and piecewise smooth initial data. An efficient time integration for these equations is based on exponential integrators in the Fourier domain. The important step in such approaches is the computation of rapidly oscillating integrals on suitably chosen contours in the complex plane using spectral methods. Several examples mainly in the linear case are discussed.

Efficient nonlinear reduced order modeling for dynamic analysis of flat structures

In the present paper, strategies for reduced order modeling of geometrically nonlinear finite element models are investigated. Simulation-free, non-intrusive approaches are considered, which do not require access to the source code of a finite element program (e.g., proprietary knowledge). Our study focus on but is not restricted to flat structures. Reduction bases are generated using bending modes and the associated modal derivatives, which span the additional subspace needed for an adequate approximation of the geometrically nonlinear response. Moreover, the reduced nonlinear restoring forces are expressed as third order polynomials in modal coordinates. Consequently, the reduced systems can be effectively solved using time-integration schemes involving only the reduced coordinates. A bottleneck in the non-intrusive methods is typically the computational effort for precomputing the polynomial coefficients and generating the reduction basis. In this regard, we demonstrate that modal derivatives have several useful properties. In particular, the modal derivatives essentially provide all the information needed for generating the polynomial coefficients for the in-plane coordinates. For condensed systems, which ignores the inertia of the in-plane modes, we show that the modal derivatives can be used effectively for recovering the in-plane displacements. Based on these findings, we propose a methodology for generating reduced order models of geometrically nonlinear flat structures in a computationally efficient manner. Moreover, we demonstrate that the concepts extend also to curved structures. The modeling techniques are validated by means of numerical examples of solid beam models and continuously supported shell models. The computational efficiency of the proposed methodology is evaluated based on the number of static evaluations needed for identifying the polynomial coefficients, as compared to the state-of-the-art methods. Furthermore, strategies for efficient time integration are discussed and evaluated.

Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear part and a nonlinear part. In the proposed methods of different orders, the responses of the linear part at the previous step are transferred by the generalized Padé approximations, and the nonlinear part’s responses of the previous step are approximated by the Gauss–Legendre quadrature together with the explicit Runge–Kutta method, where the explicit Runge–Kutta method is used to calculate function values at quadrature points. For reducing computations and rounding errors, the 2m algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10−16 (computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems.

A parallel algorithm for direct simulation of fluidized beds

AbstractThis paper presents a parallel algorithm for massive direct simulations of fluidized beds. It is based on an implicit–explicit (IMEX) technique for efficient and stable time integration of the incompressible Navier–Stokes equations in a domain that fits the boundaries of the particles at any time moment. The only modelling element in the algorithm is the model for particle–particle collisions, which is of the so‐called soft‐sphere type. The parallel performance of this technique is comparable to the performance of fully explicit methods. The numerical simulations of fluidized beds show very realistic effects that demonstrate the relevance of the simulation approach.

An efficient hybrid implicit time integration method with high accuracy for linear and nonlinear dynamics

In this work, an effective hybrid implicit method is proposed to analyze linear and nonlinear dynamic problems. The proposed hybrid method is a type of three sub-step method and is unconditionally stable. New method has multiple free parameters which can be employed to control accuracy and numerical dissipation. The new method can obtain controllable numerical dissipation which ranges from the non-dissipative case to the asymptotic annihilating case. Meanwhile, high computation accuracy than classical sub-step methods can be obtained. The new method can obtain the identical effective stiffness matrix for three sub-steps to save computational cost and internal storage for linear cases. Verified by numerical examples, the new method has simple calculation procedure, and is efficient for linear dynamics. Especially for nonlinear dynamics, the new method shows high solution accuracy and excellent energy conservation performance when compared with other methods.

Modeling dynamic brittle fracture in functionally graded materials using hyperbolic phase field and smoothed particle hydrodynamics

We present a novel particle-based computational framework for the numerical simulation of dynamic crack propagation in functionally graded materials under highly dynamic loading conditions and large deformations. The approach is based on an innovative computational method that solves phase field of brittle fracture with smoothed particle hydrodynamics. The meshfree nature of the discretization technique allows for the simulation of scenarios involving extreme deformations and material separation, as opposed to conventional mesh-based computational techniques such as the finite element method. At the same time, the damage evolution is governed by a hyperbolic partial differential equation that allows for efficient explicit time integration and avoids the complexities of solving linear systems of equations. The framework is verified and validated against other computational approaches and experimental results. Finally, the proposed approach is applied to some challenging impact scenarios that involve fast dynamics or large deformations, and it is shown that it can be easily used for identifying material gradation profiles that manipulate crack propagation.

Assessment of an adaptive time integration strategy for a high‐order discretization of the unsteady RANS equations

SummaryMany industrial applications, for example, aeronautics, aeroacoustics, and turbomachinery, are characterized by complex turbulent flow problems, whose numerical studies are mostly based on statistical models. These models rely on the Navier–Stokes equations averaged in time, namely the Reynolds‐averaged Navier–Stokes (RANS) equations. RANS unsteady solutions require accurate and efficient time integration strategies, and the choice of the time‐step can have a strong impact on the robustness and efficiency of the simulation. Adaptive time step algorithms can improve considerably the effectiveness of time integration, but in literature little information is available on their application to unsteady RANS equations for high‐Reynolds number turbulent flows. This work aims at reducing this gap, presenting a numerical investigation of the performance of different adaptive temporal strategies applied to linearly implicit Rosenbrock‐type schemes within a high‐order discontinuous Galerkin framework. Several test cases of increasing stiffness and difficulty are considered for both compressible and incompressible turbulent flows.

Coupled sliding–decohesion–compression model for a consistent description of monotonic and fatigue behavior of material interfaces

A thermodynamically consistent model proposed in this work introduces a generative symbolic framework for computational components, which can capture the inelastic behavior of interfaces with 3D kinematics in response to monotonic, cyclic, and fatigue loading histories. It contributes to a long list of existing cohesive zone models by introducing novel forms of potentials for free energy, threshold function, and dissipation to deliver the following model properties: (i) the evolution equations derived from the potentials are linked in such a way that damage develops along with the cumulative measure of plastic sliding, (ii) the convex, continuous, and smooth shape of the threshold function introduced in the effective stress domain enables a general and efficient time integration that automatically handles unloading and reloading, and (iii) the non-associative flow potential includes a scalable degree of coupling between normal and tangential damage evolution. The studies performed in the paper demonstrate that these model ingredients lead to a consistent coverage of arbitrary non-proportional loading scenarios in tangential and normal directions. For monotonic loading, the presented examples illustrate the ability of the model to reproduce the shear dilatancy, pressure sensitivity including the vertex effect, stiffness recovery in compression, and abrasion due to sliding. For cyclic fatigue loading, the studies show that the model reflects the link between the hysteretic behavior and the fatigue damage evolution. This link constitutes the basis for a physically sound description of fatigue degradation along material interfaces. It manifests itself in the presented energy breakdown with distinguished fractions of energy release due to damage and plasticity ascribed to normal and tangential directions. Two calibration and validation examples using pull-out tests exposed to monotonic and cyclic loading are provided to show the applicability of the model for the characterization of the bond between steel bars and concrete. A publicly available implementation using a computer algebra system (CAS) is provided in the form of an interactive Jupyter notebook to document the reproducibility of the results.

Core-edge 2D fluid modeling of full tokamak discharge with varying magnetic equilibrium: from WEST start-up to ramp-down

In the present work we investigate for the first time the 2D fluid transport of the plasma in WEST during an entire discharge from the start-up to the ramp-down (shot #54487). The evolution of density profile, electron and ion temperatures together with the experimental magnetic equilibrium, total current and gas-puff rate is investigated. Comparisons with the interferometry diagnostic show a remarkable overall qualitative agreement during the discharge that can be quantitative at some locations in the plasma core. If at the onset of the X-points during the ramp-up the electron heat flux is dominant at the target, present results show that the ion heat flux becomes dominant during the stationary phase of the discharge. Using a simple model for erosion, present results assess the tungsten sputtering due to deuterium ions during the start-up and ramp-up phases of the discharge and confirm the need to consider full discharge simulation to accurately treat the W source of contamination. This work also demonstrates the interest of developing magnetic equilibrium free solver including efficient time integration to step toward predictive capabilities in the future for fusion operation.

Highly precise and efficient solution strategy for linear heat conduction and structural dynamics

AbstractTime‐dependent parabolic and hyperbolic equations are widely encountered in practical engineering. Accurate and fast solution methods for such equations have always attracted attention. To reach this aim, this article proposes a strategy to construct highly precise and efficient time integration methods (TIMs) for linear heat conduction and structural dynamic systems. In the proposed strategy, an integrated amplification matrix of a TIM is created to precisely transfer the free responses of the previous step, and the Gauss–Legendre quadrature is employed to precisely compute the forced responses of the current step. To reduce computational cost and rounding error, a 2m algorithm and a method of storing incremental matrix are applied in the construction of the integrated matrix. Moreover, the L‐stable backward difference formula for heat conduction and the generalized trapezoidal rule for structural dynamics, which is A‐stable and has controllable dissipation, are utilized to conduct this strategy. Numerical experiments validate that compared with some existing TIMs, the TIMs generated by the proposed strategy enjoy advantages both in precision and efficiency.

Efficient adaptive exponential time integrators for nonlinear Schrödinger equations with nonlocal potential

The performance of exponential-based numerical integrators for the time propagation of the equations associated with the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method for the approximation of the multi-particle Schrödinger equation in one space dimension is assessed. Among the most popular integrators such as Runge–Kutta methods, time-splitting, exponential integrators and Lawson methods, exponential Lawson multistep methods with one predictor–corrector step provide the best stability and accuracy at the least effort. This assessment is based on the observation that the evaluation of the nonlocal terms associated with the potential is the computationally most demanding part of such a calculation in our setting. In addition, the predictor step provides an estimator for the local time-stepping error, thus allowing for adaptive time-stepping which reflects the smoothness of the solution and enables to reliably control the accuracy of a computation in a robust way, without the need to guess an optimal stepsize a priori. One-dimensional model examples are studied to compare different time integrators and demonstrate the successful application of our adaptive methods.

Efficient Time Integration of Nonlinear Partial Differential Equations by Means of Rosenbrock Methods

We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use a suitable choice of boundary values for the internal stages. The main difference from the linear case comes from the difficulty to calculate those boundary values exactly in terms of data. In any case, the implementation is cheap and simple since, at each stage, just some additional terms concerning those boundary values and not the whole grid must be added to what would be the standard method of lines.

Partitioned formulation of contact‐impact problems with stabilized contact constraints and reciprocal mass matrices

AbstractThis work presents an efficient and accuracy‐improved time explicit solution methodology for the simulation of contact‐impact problems with finite elements. The proposed solution process combines four different existent techniques. First, the contact constraints are modeled by a bipenalty contact‐impact formulation that incorporates stiffness and mass penalties preserving the stability limit of contact‐free problems for efficient explicit time integration. Second, a method of localized Lagrange multipliers is employed, which facilitates the partitioned governing equations for each substructure along with the completely localized contact penalty forces pertaining to each free substructure. Third, a method for the direct construction of sparse inverse mass matrices of the free bodies in contact is combined with the localized Lagrange multipliers approach. Finally, an element‐by‐element mass matrix scaling technique that allows the extension of the time integration step is adopted to improve the overall performance of the algorithm. A judicious synthesis of the four numerical techniques has resulted in an increased stable explicit step‐size that boosts the performance of the bipenalty method for contact problems. Classical contact‐impact numerical examples are used to demonstrate the effectiveness of the proposed methodology.

A Continuum Model for Simulation of Large Biological Lipid Membranes

Joint Design of Advanced Computing Solutions for Cancer (JDACS4C) - Pilot 2 Team. Presented by: Timothy S. Carpenter. Membrane proteins have important cellular roles and comprise the vast majority of drug targets. Membrane proteins interact strongly with the lipids, and protein function can be affected by local lipid compositions. To properly explore this relationship requires a bilayer that is not only of sufficient size to contain relevant compositional fluctuations, but is also run for enough time for proteins to react to both local lipid compositions and other nearby proteins. The appropriate length- and time-scales are at least microns and milliseconds. Concurrently, the mechanisms of interaction between lipids and proteins are defined by molecular structure at the sub-nanometer scale, and are typically explored by molecular dynamics simulations limited to just microseconds of time evolution.Within the JDACS4C Pilot 2 effort, we have developed the multiscale framework MuMMI for accessing ensemble averages and very long time behavior of these lipid-protein systems at atomistic levels of detail.We present the GridSim2D component of that framework: a continuum level simulation capability that can address the very long time and large-scale behavior of lipid bilayers and associated proteins. The formulation follows a dynamic density function theory approach and can be parametrized entirely from molecular dynamics simulations, making it as consistent as possible to underlying molecular forcefields. Using this approach, we can simulate a bilayer microns in diameter for hundreds of milliseconds. The implementation uses special numerical algorithms to limit communication bandwidth and allow efficient time-integration of equations of motion. It is parallelized using OpenMP and MPI and is scalable to tens of thousands of cores.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-ABS-815246.

Efficient exponential time integration for simulating nonlinear coupled oscillators

In this paper, we propose an advanced time integration technique associated with explicit exponential Rosenbrock-based methods for the simulation of large stiff systems of nonlinear coupled oscillators. In particular, a novel reformulation of these systems is introduced and a general family of efficient exponential Rosenbrock schemes for simulating the reformulated system is derived. Moreover, we show the required regularity conditions and prove the convergence of these schemes for the system of coupled oscillators. We present an efficient implementation of this new approach and discuss several applications in scientific and visual computing. The accuracy and efficiency of our approach are demonstrated through a broad spectrum of numerical examples, including a nonlinear Fermi–Pasta–Ulam–Tsingou model, elastic and nonelastic deformations as well as collision scenarios focusing on relevant aspects such as stability and energy conservation, large numerical stiffness, high fidelity, and visual accuracy.

Application of scaled boundary finite element method for vibration-based structural health monitoring of breathing cracks

The dynamic response of the host structure to a high-frequency actuation is usually used for the detection of tiny damage in structures in the form of breathing crack. The simulation of the microcrack’s effect on the response is essential for several damage identification targets. The conventional finite element method suffers from very small mesh size requirements to address the high-frequency problems, resulting in very large mass and stiffness matrices. In this study, the scaled boundary finite element method was applied to model different schemes of structural health monitoring of a structure with breathing cracks based on high-frequency vibration. The scaled boundary finite element method discretizes only the boundary of the model and thus substantially reduces the size of structural matrices. The node-to-node contact strategy was introduced to the scaled boundary finite element method to capture the contact problem that occurs during the vibration of the breathing crack. As breathing crack vibration results in some nonlinear effects, the simulation of three phenomena was of interest: higher harmonic generation, frequency shift, and vibro-acoustic modulation. A shooting method was used for efficient time integration and description of the frequency response function in the nonlinear regime. According to the results, the scaled boundary finite element method is of great power, efficiency, and accuracy to treat the contact problems, especially in high-frequency regimes. Moreover, the nonlinear methods provide certain advantages over the linear techniques in the early detection of incipient damage.