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Article Dans Une Revue Finite Elements in Analysis and Design Année : 2007

Stability of semidiscrete formulations for elastodynamics at small time steps

Eran Grosu
  • Fonction : Auteur
Isaac Harari
  • Fonction : Auteur

Résumé

Solutions of direct time-integration schemes for elastodynamics that converge in time to conventional semidiscrete formulations may be polluted at small time steps by spurious oscillations that violate the principle of causality, for example by arising before wave fronts. This degradation is not an artifact of the time-marching scheme, but rather a property of the solution of the semidiscrete formulation itself. An analogy to singularly perturbed elliptic problems provides an upper bound on the time step for the onset of these oscillations. A simple procedure of spatial stabilization is proposed to remove this pathology from implicit time-integration schemes, without affecting unconditional temporal stability. Spatially stabilized implicit time-integration methods are free of noncausal oscillations at small time steps. 1. Introduction Dynamic behavior of elastic bodies (elastodynamics) is described by the equation of motion, which is continuous in space and time. Finite element methods are usually used to obtain numerical solutions for problems in structural dynamics. The common approach to transient computation (the method of lines [1]) is time-integration of a semidiscrete formulation, which is obtained by spatial discretization. Thus, the approximation is carried out in two stages. First, spatial approximation , e.g., by standard finite element methods, leads to the semidiscrete formulation (a coupled system of ordinary differential equations in time). Then, temporal discretization by time-integration schemes results in a system of algebraic equations at each time level. Perhaps the most widely used family of direct methods for solving the semidiscrete equation of motion is the Newmark family of methods for time-integration. The unconditionally stable in time, second-order accurate trapezoidal rule is, probably, the most commonly used implicit algorithm among the Newmark family.
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Dates et versions

hal-01634271 , version 1 (13-11-2017)

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Eran Grosu, Isaac Harari. Stability of semidiscrete formulations for elastodynamics at small time steps. Finite Elements in Analysis and Design, 2007, 43 (6-7), pp.533 - 542. ⟨10.1016/j.finel.2006.12.006⟩. ⟨hal-01634271⟩
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