A re‐evaluation of overshooting in time integration schemes: The neglected effect of physical damping in the starting procedure

Self Cite

This article introduces a new additional test concept and numerical property of implicit time integration methods termed as “aliasing” of the external load which must be considered during the algorithm design process. Where most numerical properties of such algorithms discussed in the literature pertain to the homogeneous components, aliasing reflects the tendency of the nonhomogeneous components to introduce errors into the solution in the asymptotic high‐frequency limit. An analytical technique is proposed to detect and quantify aliasing, defined in terms of the resulting representative solution. A theorem establishes whether a representative solution with aliasing can be bounded up to a constant by the external force values and thus treated as acceptable if the bound is sufficiently small. For the U0 family of algorithms within the GSSSS‐II framework containing several traditional and closely related methods, this is not an issue. However, the numerically dissipative GSSSS‐II V0 family of methods is shown to exhibit bounded aliasing with a constant roughly equal to 2, while the “structure‐dependent” Chang Family Method is shown to exhibit unbounded aliasing, which is not acceptable for practical dynamic computations. Numerical verifications are performed with single‐degree and multi‐degree of freedom problems with a series of selected algorithms which exhibit either no aliasing, tightly bounded aliasing, unbounded aliasing, or large bounded aliasing, and practical recommendations are given for acceptable aliasing of the load with special emphasis on MDOF problems such as those arising from finite element discretizations.

Section: Numerical Properties Of Single-step Methodsmentioning

confidence: 99%

Ω

.…”

Section: Numerical Properties Of Single‐step Methodsmentioning

confidence: 99%

“…This illustrates that additional care must be taken in the design of algorithms with respect to overshooting. This is covered in greater detail in Reference 7.…”

Section: Overshootingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Load aliasing—A new additional test concept for effective control of nonhomogeneous high‐frequency behavior in linear multistep methods

Maxam

Tamma

2022

Self Cite

Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

Zhao

et al. 2023

SummaryThis paper reviews the published composite three‐sub‐step implicit algorithms all of which adopt the trapezoidal rule in the first sub‐step. Three optimal families of three‐sub‐step implicit algorithms are developed to achieve second, third, and fourth‐order accuracy. The present second‐ and third‐order methods achieve identical effective stiffness matrices, thus embedding optimal spectral characteristics. Besides, both of them impose two dissipative parameters ( and ) to control numerical dissipation imposed in the second and third sub‐steps. The parameter adjusts overall numerical dissipation in the whole integration schemes, while the firstly used parameter can change numerical dissipation in the second sub‐step. The numerical example has shown the superiority of controlling middle sub‐step dissipation via . The present fourth‐order method is moderately dissipative due to achieving higher‐order accuracy, but it presents a more reasonable sub‐step division than the published fourth‐order three‐sub‐step trapezoidal rule. Linear and nonlinear examples are simulated to show the numerical performance and superiority of the three novel methods. This paper recommends using the proposed second‐ and third‐order three‐sub‐step methods to solve various dynamic problems.

On designing and developing single‐step second‐order implicit methods with dissipation control and zero‐order overshoots via subsidiary variables

Lian

et al. 2023

Hilber and Hughes gave several competitive demands on the implicit methods in 1978. However, there are no such integration methods in the traditional algorithm design. Via imposing subsidiary variables, this article proposes two novel implicit methods to splendidly achieve these competitive demands without increasing computational costs. The two novel methods are self‐starting, unconditionally stable, single‐solve, identically second‐order accurate, controllably dissipative, and non‐overshooting. The novel methods achieve the maximal dissipation of auxiliary variables in the high‐frequency limit, although such the design is not a must. The first novel method uses auxiliary velocity and acceleration variables, whereas the second one employs two auxiliary acceleration variables. After embedding desired numerical characteristics, the novel methods employ four eigenvalues of amplification matrices in the high‐frequency limit as user‐specified parameters to flexibly control numerical high‐frequency dissipation. To reduce the number of use‐specified parameters, this article gives some recommended sub‐families of algorithms, such as optimal low‐frequency dissipation schemes. The article further refines the error analysis to analytically calculate amplitude and phase errors for some implicit methods. The error analysis technique can be applied to almost all direct time integration methods. Comparisons of amplitude and phase errors highlight the superiority of the novel methods over the published algorithms. In addition, some classical measures, such as numerical damping ratios and relative period errors, are analyzed and compared. Numerical examples are solved to show the novel methods' superiority.