Efficient high-order accurate explicit time-marching procedures for dynamic analyses
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ORIGINAL ARTICLE
Efficient high‑order accurate explicit time‑marching procedures for dynamic analyses Delfim Soares Jr.1 Received: 7 August 2020 / Accepted: 23 September 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, η-order accurate truly-explicit time-marching procedures are proposed for dynamic analyses. In these new techniques, just η–1 stiffness matrix–vector operations are required for solution, per time-step, providing more efficient approaches than standard high-order time-marching procedures, which demand η matrix–vector operations per time-step. In addition, as illustrated along this work, the novel formulations are also more accurate than equivalent standard techniques, providing very effective time-domain solution procedures. Three truly-explicit time-marching procedures are discussed here, describing second-, third- and fourth-order accurate methodologies. Keywords Explicit analysis · High-order techniques · Accuracy · Efficiency
1 Introduction Time-domain hyperbolic equations have considerable applicability and relevance in various branches of engineering and science. Since it is very difficult to evaluate exact solutions for these transient equations, numerical approaches are usually employed to find approximate responses, and step-by-step time integration techniques are commonly applied when dynamic problems are regarded, because of their numerous advantages analysing a great deal of initial value models [1–33]. In this work, three novel truly-explicit time-marching procedures are proposed for dynamic analyses. As it is well-known, in explicit approaches, effective matrices are formulated without considering stiffness matrices in their compositions, allowing eliminating solvers procedures when lumped mass and damping matrices are regarded. In trulyexplicit formulations, the effective matrix of a model is solely represented by its mass matrix, allowing introducing non-diagonal damping matrices into the analyses without considering solver routines for solution. The three trulyexplicit algorithms that are proposed here provide second-, third-, and fourth-order accurate techniques, thus, each * Delfim Soares Jr. [email protected] 1
Structural Engineering Department, Federal University of Juiz de Fora, Juiz de Fora, MG 36036‑330, Brazil
approach may be applied according to the requested order of accuracy. The novel second-order accurate technique is spectrally equivalent to the standard central difference (CD) method, once undamped models are regarded. Nevertheless, the new approach stands as a truly self-starting formulation (all three novel techniques are truly self-starting), providing an extra positive feature that is not followed by the CD. For damped models, the new second-order technique becomes more accurate than the CD, further illustrating additional advantages for the new approach (besides being truly explicit and self-starting). The proposed third- and fourth-order accurate techniques also provide more accurate results than their equ
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