A variable step-size selection method for implicit integration schemes

Holsapple, Raymond; Iyer, Ram; Doman, David B.

doi:10.1109/acc.2006.1657179

Cited by 9 publications

(17 citation statements)

References 14 publications

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“…1) solution using the following approach: 1) Obtain a number of numerical solutions using fixed-step solver for a range of solver's orders and sampling times, in such a way, that the solutions are expected to be stable and accurate, e.g. by arbitrarily specifying the maximum step size [26,27]. 2) Test if the solutions converge with an increasing solver order.…”

Section: Fig 1 Comparison Of Exact and Numerical Solutionsmentioning

confidence: 99%

The Analysis of Vehicle’s In-Flight Behaviour Using Quasi-LPV and Nonlinear Models

Machala

Dobre

Albisser

et al. 2019

AIAA Scitech 2019 Forum

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A vehicle's in-flight behaviour can be represented by the Newton-Euler equations of motion: usually, such a model has a nonlinear and continuous description based on ordinary differential equations. The model structure can be altered using analytic transformations and, when the model is implemented, its numerical precision may depend on the selection of a numerical solver. Hence, the selection of a model's structure and an appropriate numerical solver can become the key development issues, if the model's numerical stability, convergence, and computational complexity are to be improved. This paper assesses the influence of a model's reference frame and a numerical solver on the accuracy of calculating a projectile's trajectory, for a case study of free-flight long-range ballistic experiments. The analysis is based on two model structures: the first is a six-degree-of-freedom nonlinear description of a spin-stabilized projectile, expressed in a rolling reference frame; the second is a quasi-LPV reformulation of the same projectile model, but expressed in a non-rolling reference frame. It is shown that inappropriate selection of a numerical solver can hinder the accuracy of the nonlinear model. At the same time, the model represented in a non-rolling reference frame offers a solution with higher accuracy, better convergence properties, and significantly reduced computation time.

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Section: Fig 1 Comparison Of Exact and Numerical Solutionsmentioning

confidence: 99%

The Analysis of Vehicle’s In-Flight Behaviour Using Quasi-LPV and Nonlinear Models

Machala

Dobre

Albisser

et al. 2019

AIAA Scitech 2019 Forum

View full text Add to dashboard Cite

show abstract

“…That means that one or several evaluations per step are carried out to achieve a local error respecting the required tolerance. The classical adaptive step-size algorithm [28] described above presents the drawback of a lack of step-size smooth progressions and, in some cases, leads to a considerable number of step rejections. This method can be improved by including a control theory approach in the expression of the step-size selection (14).…”

Section: 22mentioning

confidence: 99%

Efficient multirate simulation techniques for multi‐physics systems with different time scales: application on an all‐electric ferry design

Hmam

Olivier

Bourguet

et al. 2017

IET Electrical Systems in Transportation

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This study introduces a multirate method for the simulation of multi-physics systems containing a wide range of time scales. This method has been designed for strongly coupled systems and is able to deal with high mutual dependency between fast and slow state variables. The proposed method based on a cycle formulation approach is applied to the ageing behaviour simulation of the energy storage unit (ESU) of an all-electric ferry. The objective is to design this ESU taking into account the electrical and thermal models, permitting to predict its ageing over the 20 years of the ferry operation. So, simulations could be excessively time consuming. The main objective behind this study is to reduce the simulation time to make possible an optimisation process. Using the proposed multirate method based on embedded extrapolation methods formulated on cycles, a speed-up factor of about 1000 compared with standard integration methods is obtained. Numerical results for the electric ferry example are presented for different tolerances in order to demonstrate the performance of the method.

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“…Step size selection is an important criterion required in solving stiff differential equations using the integration method, [1]. It is however important to state that too small or too large a step size affects the efficiency of any integration method.…”

Section: Introductionmentioning

confidence: 99%

Variable Step Hybrid Block Method for the Approximation of Kepler Problem

2022

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In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps, minimize the number of formulae stored in the code, and ensure that the method is zero-stable. The method is formulated by integrating the Lagrange polynomial with limits of integration selected at special points. The article further analyzed the stability, order, consistency, and convergence properties of the VSHBM. The stability regions of the VSHBM at different values of the step size ratios were also plotted and plots showed that the method is fit for solving the Kepler problem. The results generated were then compared with some existing methods, including the MATLAB inbuilt stiff solver (ode 15 s), with respect to total number of failure steps, total number of steps, total function calls, maximum error, and computation time.

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A variable step-size selection method for implicit integration schemes

Cited by 9 publications

References 14 publications

The Analysis of Vehicle’s In-Flight Behaviour Using Quasi-LPV and Nonlinear Models

The Analysis of Vehicle’s In-Flight Behaviour Using Quasi-LPV and Nonlinear Models

Efficient multirate simulation techniques for multi‐physics systems with different time scales: application on an all‐electric ferry design

Variable Step Hybrid Block Method for the Approximation of Kepler Problem

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