Integration Step Size Research Articles

A second-order accurate three sub-step composite algorithm for structural dynamics

In this paper, a novel three sub-step composite algorithm with desired numerical properties is developed. The proposed method is a self-starting, unconditionally stable and second-order accurate implicit algorithm without overshoot. Particularly, the second-order accuracy in time is achieved in its final form, but it is not required in each sub-step. Its unique algorithmic parameter is analyzed to achieve the unconditional stability and it shares the identical effective stiffness matrix inside three sub-steps to save the computational cost in linear analyses. The same as the Bathe algorithm, the proposed algorithm is always L-stable, meaning that the spurious high-frequency modes can be effectively eliminated. Three numerical examples are simulated to illustrate the superiority of the proposed algorithm over some existing implicit algorithms. The first numerical simulation, solving a linear single-degree-of-freedom system, shows less period elongation errors and the second-order accuracy of the present scheme. The second one, a clamped-free bar excited by the end load, shows the ability of effectively damping out the unexpected high-frequency modes. The last example solves the nonlinear mass-spring system with variable degree-of-freedoms and illustrates that the composite sub-step algorithm can save more computational cost than the traditional implicit algorithm when the integration step size is selected appropriately.

Explicit time integration of the stiff chemical Langevin equations using computational singular perturbation.

A stable explicit time-scale splitting algorithm for stiff chemical Langevin equations (CLEs) is developed, based on the concept of computational singular perturbation. The drift term of the CLE is projected onto basis vectors that span the fast and slow subdomains. The corresponding fast modes exhaust quickly, in the mean sense, and the system state then evolves, with a mean drift controlled by slow modes, on a random manifold. The drift-driven time evolution of the state due to fast exhausted modes is modeled algebraically as an exponential decay process, while that due to slow drift modes and diffusional processes is integrated explicitly. This allows time integration step sizes much larger than those required by typical explicit numerical methods for stiff stochastic differential equations. The algorithm is motivated and discussed, and extensive numerical experiments are conducted to illustrate its accuracy and stability with a number of model systems.

An implementation of geometric integration within Matlab

Geometric integrators allow preservation of specific geometric properties of the exact flow of differential equation systems, such as energy, phase-space volume, and time-reversal symmetry, and are particularly reliable for long-run integration. In this study, variable step size composition methods and Gauss methods are implemented in Matlab library integrators, and are tested with several representative problems, including the Kepler problem, the outer solar system and a conservative Lotka–Volterra system. Variable step size integrators often perform better than their fixed step size counterparts and the numerical results show excellent long time preservation of the Hamiltonian in these examples.

A Line Integral Convolution Method With Dynamically Determining Step Size and Interpolation Mode for Vector Field Visualization

The line integral convolution method can obtain a continuous and dense vector streamline texture. The traditional fixed step size gets larger texture noise, while the exact calculation step method (such as the Runge-Kutta method) is time-consuming. In the process of generating streamline texture, if using bilinear interpolation calculation, we can get the method of calculating the point vector value, and it is also time-consuming. To address the above-mentioned problems, the line integral convolution method for dynamically determining integral step size and interpolation mode is proposed. The effect of streamline drawing mainly depends on the size of the integration step and the interpolation mode. The method proposed in this paper gives the rules for determining the integration step size and interpolation mode according to the vector speed size and the angle between two vectors. The experimental results show that compared with the previous fixed step method and the bilinear interpolation method, the proposed method not only has faster computation speed but also clearer texture and stronger contrast.

Advances in numerical methods under wiggler period averaging for free electron laser simulation

Advances in numerical methods for free-electron-laser (FEL) simulation under wiggler period averaging (WPA) are presented. First, WPA is generalized using the perturbative Lie map method. The conventional WPA is identified as the leading order contribution. Next, the shot-noise model under WPA is improved along with a particle migration scheme across the numerical mesh. The artificial shot noise arising from particle migration is suppressed. The improved model also allows using arbitrary mesh size, slippage resolution, and integration step size. These advances will improve modeling of longitudinal beam profile evolution for fast FEL simulation.

Huxley-type cross-bridge models in largeish-scale musculoskeletal models; an evaluation of computational cost

A Huxley-type cross-bridge model is attractive because it is inspired by our current understanding of the processes underlying muscle contraction, and because it provides a unified description of muscle's mechanical behavior and metabolic energy expenditure. In this study, we determined the computational cost for task optimization of a largeish-scale musculoskeletal model in which muscles are represented by a 2-state Huxley-type cross-bridge model. Parameter values defining the rate functions of the Huxley-type cross-bridge model could be chosen such that the steady-state force-velocity relation resembled that of a Hill-type model. Using these parameter values, maximum-height squat jumping was used as the example task to evaluate the computational cost of task optimization for a skeletal model driven by a Huxley-type cross-bridge model. The optimal solutions for the Huxley- and Hill-type muscle models were similar for all mechanical variables considered. Computational cost of the Huxley-type cross-bridge model was much higher than that of the Hill-type model. Compared to the Hill-type model, the number of state variables per muscle was large (2 vs about 18,000), the integration step size had to be about 100 times smaller, and the computational cost per integration step was about 100 times higher.

Stability and bifurcation analysis of a discrete predator\u2013prey system with modified Holling\u2013Tanner functional response

In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.

Rational hybrid Monte Carlo with block solvers and multiple pseudofermions

The dominant cost of most lattice QCD simulations is the inversion of the Dirac operator required to calculate the force term in the RHMC update. One way to improve this situation is to use multiple pseudofermions, which reduces the size and variance of this force and hence allows a larger integration step size to be used. This means fewer force term calculations are required, but at the cost of having to invert the Dirac operator for each pseudofermion field. This bottleneck can be addressed: recently there has been renewed interest in the use of block Krylov solvers, which can solve multiple right hand side vectors with significantly fewer iterations than are required if each vector is solved using a separate Krylov solver. We combine these two ideas, achieving a significant speed-up of RHMC lattice QCD simulations.

The lower bound error as an auxiliary technique to select the integration step-size in the simulation of chaotic systems

The lower bound error as an auxiliary technique to select the integration step-size in the simulation of chaotic systems

Computationally Efficient Non-linear Electro-hydraulic Actuation System Model for Real-time Simulation

An electro-hydraulic actuation system with critically lapped servovalve and pressurized return line is exposited. The dynamic equations governing the behavior of various components are explained; using these nonlinear equations, a simulation model is developed in Simulink® software for the purpose of controller design and verification. Model requires very small integration step size as the equations involved are stiff. A relationship between the various pressures in the system is derived, and a simplified method for the nonlinear modeling of electro hydraulic actuating system is developed using these relationships. The simplified model is computationally efficient as the integration step-size is large compared to the original model. The computationally efficient model finds application in real-time hardware-in-loop simulation (HILS).

A modified homogenized highly precise direct integration method for time-varying nonhomogeneous systems

This article investigates a novel method for time-varying systems with an impact term; we call it a modified homogenized highly precise direct integration method. Modified homogenized highly precise direct integration method can deal with the time-varying nonhomogeneous systems effectively by direct integration with high precision. Even though it is often difficult to select the time step size of integration properly for stiff problems, modified homogenized highly precise direct integration method can effectively deal with the this problem with a large time step size. By introducing new variants twice, modified homogenized highly precise direct integration method can easily deal with the nonhomogeneous term by a novel way, inherit the advantages of highly precise direct integration method and avoid the calculation of inverse matrix. The convergency and efficiency analyses are given in this article. Several numerical simulations and an application example are presented to demonstrate the high accuracy, eff...

Compliant contact versus rigid contact: A comparison in the context of granular dynamics.

We summarize and numerically compare two approaches for modeling and simulating the dynamics of dry granular matter. The first one, the discrete-element method via penalty (DEM-P), is commonly used in the soft matter physics and geomechanics communities; it can be traced back to the work of Cundall and Strack [P.Cundall, Proc. Symp. ISRM, Nancy, France 1, 129 (1971); P. Cundall and O. Strack, Geotechnique 29, 47 (1979)GTNQA80016-850510.1680/geot.1979.29.1.47]. The second approach, the discrete-element method via complementarity (DEM-C), considers the grains perfectly rigid and enforces nonpenetration via complementarity conditions; it is commonly used in robotics and computer graphics applications and had two strong promoters in Moreau and Jean [J. J. Moreau, in Nonsmooth Mechanics and Applications, edited by J. J. Moreau and P. D. Panagiotopoulos (Springer, Berlin, 1988), pp. 1-82; J. J. Moreau and M. Jean, Proceedings of the Third Biennial Joint Conference on Engineering Systems and Analysis, Montpellier, France, 1996, pp.201-208]. The DEM-P and DEM-C are manifestly unlike each other: They use different (i) approaches to model the frictional contact problem, (ii) sets of model parameters to capture the physics of interest, and (iii) classes of numerical methods to solve the differential equations that govern the dynamics of the granular material. Herein, we report numerical results for five experiments: shock wave propagation, cone penetration, direct shear, triaxial loading, and hopper flow, which we use to compare the DEM-P and DEM-C solutions. This exercise helps us reach two conclusions. First, both the DEM-P and DEM-C are predictive, i.e., they predict well the macroscale emergent behavior by capturing the dynamics at the microscale. Second, there are classes of problems for which one of the methods has an advantage. Unlike the DEM-P, the DEM-C cannot capture shock-wave propagation through granular media. However, the DEM-C is proficient at handling arbitrary grain geometries and solves, at large integration step sizes, smaller problems, i.e., containing thousands of elements, very effectively. The DEM-P vs DEM-C comparison is carried out using a public-domain, open-source software package; the models used are available online.

Modular representation of asynchronous geometric integrators with support for dynamic topology

Geometric integrators play an essential role for simulating second-order energy preserving systems, offering an alternative to the decomposition of systems into first-order Ordinary Differential Equations. This approach, although commonly used nowadays in modeling and simulation software, is not acceptable when long simulation runs are required. In this work we develop a modular representation of geometric, adaptive step-size integrators using the Heterogeneous Flow Systems Specification (HyFlow) formalism. Modularity is achieved in HyFlow through the use of an explicit definition of sampling that is treated as a first-order construct, enabling a novel representation of continuous systems and their seamless integration. We show that the HyFlow representation enables the interoperability of geometrical integrators with other families of models including, for example, conventional integrators, enhancing the ability to represent complex systems. HyFlow sampling enables geometric integrators to operate asynchronously, contributing to simulation efficiency by allowing the sampling rate to de defined independently by each component. We demonstrate that HyFlow-based geometric integrators can be used to model systems with a dynamic topology. In addition, we show that the modifying model topology at run-time can provide an effective solution to some systems exhibiting Zeno behavior.

BIFURCATION AND CHAOS IN A DISCRETE TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE WITH GENERALIZED HOLLING TYPE Ⅲ FUNCTIONAL RESPONSE

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type Ⅲ functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R₊². Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

Bifurcation Analysis and Chaos Control in a Discrete-Time Parasite-Host Model

A discrete-time parasite-host system with bifurcation is investigated in detail in this paper. The existence and stability of nonnegative fixed points are explored and the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. And we also prove the chaos in the sense of Marotto. The numerical simulations not only illustrate the consistence with the theoretical analysis, but also exhibit other complex dynamical behaviors, such as bifurcation diagrams, Maximum Lyapunov exponents, and phase portraits. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period orbits, attracting invariant cycles and chaotic attractors of the discrete-time parasite-host system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point by using the feedback control method.

Efficient Integration of Coupled Electrical-Chemical Systems in Multiscale Neuronal Simulations.

Multiscale modeling and simulations in neuroscience is gaining scientific attention due to its growing importance and unexplored capabilities. For instance, it can help to acquire better understanding of biological phenomena that have important features at multiple scales of time and space. This includes synaptic plasticity, memory formation and modulation, homeostasis. There are several ways to organize multiscale simulations depending on the scientific problem and the system to be modeled. One of the possibilities is to simulate different components of a multiscale system simultaneously and exchange data when required. The latter may become a challenging task for several reasons. First, the components of a multiscale system usually span different spatial and temporal scales, such that rigorous analysis of possible coupling solutions is required. Then, the components can be defined by different mathematical formalisms. For certain classes of problems a number of coupling mechanisms have been proposed and successfully used. However, a strict mathematical theory is missing in many cases. Recent work in the field has not so far investigated artifacts that may arise during coupled integration of different approximation methods. Moreover, in neuroscience, the coupling of widely used numerical fixed step size solvers may lead to unexpected inefficiency. In this paper we address the question of possible numerical artifacts that can arise during the integration of a coupled system. We develop an efficient strategy to couple the components comprising a multiscale test problem in neuroscience. We introduce an efficient coupling method based on the second-order backward differentiation formula (BDF2) numerical approximation. The method uses an adaptive step size integration with an error estimation proposed by Skelboe (2000). The method shows a significant advantage over conventional fixed step size solvers used in neuroscience for similar problems. We explore different coupling strategies that define the organization of computations between system components. We study the importance of an appropriate approximation of exchanged variables during the simulation. The analysis shows a substantial impact of these aspects on the solution accuracy in the application to our multiscale neuroscientific test problem. We believe that the ideas presented in the paper may essentially contribute to the development of a robust and efficient framework for multiscale brain modeling and simulations in neuroscience.

About development of parallel difference schemes modeling with variation steps in the settlement block

Offered the parallel difference schemes for modeling of dynamic objects that allow you to control the step size in numerical integration. Calculation schemes are formed on the basis of collocation multistep block methods. Numerical solutions for each calculated block implemented using Newtonian iterations. Figs.: 4. Refs.: 12 titles. Keywords: difference scheme; control the step size; collocation multistep block method; Newton iteration.

Adaptive time stepping in pseudo-dynamic testing of earthquake driven structures

The problem of assessing errors in implementing time-marching algorithms in the context of pseudo-dynamic seismic testing of structures is considered. These errors occur in implementing the numerical and experimental steps of the test procedure. The study investigates how a linearized variational equation can be augmented with the governing equation of motion to track the effect of the errors, and, accordingly, adjust the step size of integration adaptively to keep a global error norm within specified limits. The governing augmented equations are integrated using an explicit operator splitting scheme. Additional efforts, in terms of evaluation of the tangent stiffness matrix, are shown to become necessary while modelling the errors. Illustrative examples include numerical studies on a set of nonlinear systems and an experimental study on a geometrically nonlinear two-storied building frame. The experimental results from pseudo-dynamic test are shown to compare reasonably well with pertinent results from an effective force test.

Numerically efficient modified Runge–Kutta solver for fatigue crack growth analysis

We present a modified Runge–Kutta algorithm which yields a conservative estimate of the crack size for fatigue crack growth even for large integration step sizes. Conservative in this context means to overestimate the crack size. Commonly used algorithms (e.g. Euler, Runge–Kutta) usually underestimate the crack growth and only converge for small step sizes to an accurate value. As the presented algorithm overestimates the crack growth even for large and converges for small integration step sizes, it can be used for instance in Monte-Carlo based probabilistic fracture mechanics simulations, which might be otherwise computational impractical.

Bifurcation analysis of a discrete-time ratio-dependent predator–prey model with Allee Effect

A discrete-time predator–prey model with Allee effect is investigated in this paper. We consider the strong and the weak Allee effect (the population growth rate is negative and positive at low population density, respectively). From the stability analysis and the bifurcation diagrams, we get that the model with Allee effect (strong or weak) growth function and the model with logistic growth function have somewhat similar bifurcation structures. If the predator growth rate is smaller than its death rate, two species cannot coexist due to having no interior fixed points. When the predator growth rate is greater than its death rate and other parameters are fixed, the model can have two interior fixed points. One is always unstable, and the stability of the other is determined by the integral step size, which decides the species coexistence or not in some extent. If we increase the value of the integral step size, then the bifurcated period doubled orbits or invariant circle orbits may arise. So the numbers of the prey and the predator deviate from one stable state and then circulate along the period orbits or quasi-period orbits. When the integral step size is increased to a critical value, chaotic orbits may appear with many uncertain period-windows, which means that the numbers of prey and predator will be chaotic. In terms of bifurcation diagrams and phase portraits, we know that the complexity degree of the model with strong Allee effect decreases, which is related to the fact that the persistence of species can be determined by the initial species densities.