Extension of LMS formulations for L‐stable optimal integration methods with U0–V0 overshoot properties in structural dynamics: the level‐symmetric (LS) integration methods

Abstract

SUMMARYIn the present article a new family of linear multi-step (LMS) optimal integration methods for stiff structural dynamic problems is synthesized. ) of design of optimal and controllable dissipation, extensions are made in this paper to a class of optimal integration methods with maximum possible damping of high frequencies separately from the degree of damping of low frequencies. Generalized single step single solve (GSSSS) optimal algorithm recently developed and basic well-known 'weighted-residual' methods are compared with 'level-symmetric' (LS) integration methods proposed. These LS methods are created as symmetric variants of extended three-level (3L-LMS) integration algorithm with direct use of dynamic equations to obtain algorithmically simple integration methods, which belong to [U0-V0] L-stable optimal class without overshoots and have the maximum damping of high-frequency modes. General formulas for L-stable family of multi-step LS-N methods are obtained. Standard two-level representation (2L-LMS) of 3L-LMS integration algorithm is also obtained, and L-stable second-order accurate LS-BDF integration method for the stiff first-order ordinary differential equations is proposed. Roots, dissipation and dispersion properties of LS-1 integration method (second-order accurate in displacement) and of other obtained LS-2, LS-3, LS-4 methods (third-order accurate in displacement) are analysed and demonstrated. Comparison with some up-to-date integration methods is considered in three numerical examples.

An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

“…The above equation is different from (72) in terms of the second integral equation in (76). Therefore, the principle of virtual work, which is called the weak form in contrast to the strong form given by (70), implicitly contains both natural and essential boundary conditions as well as the governing equation of motion.…”

“…It is not suitable for the admissible functions to be applied to the above equation due to the continuity requirements. The continuity requirement for the admissible functions to be applied can be relaxed in (75) [30,43,63,102,103,106], because (75) is weaker than (72) or (76) in the sense of continuity requirements for the displacement-based finite element method. Therefore, it is quite natural to employ (75) to discretize the space.…”

“…Therefore, the principle of virtual work, which is called the weak form in contrast to the strong form given by (70), implicitly contains both natural and essential boundary conditions as well as the governing equation of motion. The modified Bubnov-Galerkin weighted residual method [88] uses (75) to discretize the space rather than (72). It is not suitable for the admissible functions to be applied to the above equation due to the continuity requirements.…”

“…Descriptive Scalar Function, the Lagrangian The BubnovGalerkin weighted residual form is given by the inner product of Cauchy's equation of motion and the virtual displacement as discussed in (72). By use of Gauss's divergence theorem and integration by parts, the Bubnov-Galerkin weighted residual form gives a weak form in (75).…”

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

“…The "generalized  -method in structural dynamics" is presented in reference [12] with "user controlled numerical dissipation". Extensions of LMS methods to L -stable optimal integration methods with 0 u -0 v overshoot properties in structural dynamics are given in reference [13]. This approach leads to level-symmetric ( LS ) integration methods.…”