Time Step Estimates for Reciprocal Mass Matrices Using Ostrowski's Bounds

Tkachuk, Anton; Kolman, Radek; González, José Luís Galán; Bischoff, Manfred; Kopačka, Ján

doi:10.7712/120119.6956.18956

Cited by 1 publication

(3 citation statements)

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“…The paper indicates that the Ostrowski's bound is the sharpest bound for most of the cases. Further study discovers that in the presence of mixed degrees of freedom, like displacement and rotations in a Bernoulli beam element, the Gershgorin's and Ostrowski's bounds substantially underestimate the critical time step 38 . Therefore, an additional similarity transformation by Fan 39 should assist the bounds.…”

Section: Introductionmentioning

confidence: 99%

“…Further study discovers that in the presence of mixed degrees of freedom, like displacement and rotations in a Bernoulli beam element, the Gershgorin's and Ostrowski's bounds substantially underestimate the critical time step. 38 Therefore, an additional similarity transformation by Fan 39 should assist the bounds. Such an estimator is tested for a 3-node plate element and a stiffened panel structure with a regular mesh and an identical similarity transformation matrix at each node.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, an additional similarity transformation by Fan 39 should assist the bounds. Such an estimator is tested for a 3‐node plate element and a stiffened panel structure with a regular mesh and an identical similarity transformation matrix at each node 38 . This opens a perspective of adjusting of this estimator to the case of elements with Allman's rotations and testing whether this estimator satisfies following requirements denoted with T* to be conservative, to be efficient for finite elements with displacement and Allman's rotations, and to provide satisfactory results for irregular and distorted meshes. …”

Section: Introductionmentioning

confidence: 99%

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Reciprocal mass matrices and a feasible time step estimator for finite elements with Allman's rotations

Tkachuk

2020

Int J Numer Methods Eng

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Finite elements with Allman's rotations provide good computational efficiency for explicit codes exhibiting less locking than linear elements and lower computational cost than quadratic finite elements. One way to further raise their efficiency is to increase the feasible time step or increase the accuracy of the lowest eigenfrequencies via reciprocal mass matrices. This article presents a formulation for variationally scaled reciprocal mass matrices and an efficient estimator for the feasible time step for finite elements with Allman's rotations. These developments take special care of two core features of such elements: existence of spurious zero-energy rotation modes implying the incompleteness of the ansatz spaces, and the presence of mixed-dimensional degrees of freedom. The former feature excludes construction of dual bases used in the standard variational derivation of reciprocal mass matrices. The latter feature destroys the efficiency of the existing nodal-based time step estimators stemming from the Gershgorin's eigenvalue bound. Finally, the developments are tested for standard benchmarks and triangular, quadrilateral, and tetrahedral finite elements with Allman's rotations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%