Open Access

ARTICLE

Mass-Stiffness Templates for Cubic Structural Elements

Carlos A. Felippa^*

Department of Aerospace Engineering Sciences and Aerospace Mechanics Research Center, University of Colorado at Boulder, Boulder, CO 80309-0429, USA

* Corresponding Author:Carlos A. Felippa. Email:

(This article belongs to the Special Issue: Advances in Computational Mechanics and Optimization
To celebrate the 95th birthday of Professor Karl Stark Pister)

Computer Modeling in Engineering & Sciences 2021, 129(3), 1209-1241. https://doi.org/10.32604/cmes.2021.016803

Received 28 March 2021; Accepted 04 June 2021; Issue published 25 November 2021

Abstract

This paper considers Lagrangian finite elements for structural dynamics constructed with cubic displacement shape functions. The method of templates is used to investigate the construction of accurate mass-stiffness pairs. This method introduces free parameters that can be adjusted to customize elements according to accuracy and rank-sufficiency criteria. One- and two-dimensional Lagrangian cubic elements with only translational degrees of freedom (DOF) carry two additional nodes on each side, herein called side nodes or SN. Although usually placed at the third-points, the SN location may be adjusted within geometric limits. The adjustment effect is studied in detail using symbolic computations for a bar element. The best SN location is taken to be that producing accurate approximation to the lowest natural frequencies of the continuum model. Optimality is investigated through Fourier analysis of the propagation of plane waves over a regular infinite lattice of bar elements. Focus is placed on the acoustic branch of the frequency-vs.-wavenumber dispersion diagram. It is found that dispersion results using the fully integrated consistent mass matrix (CMM) are independent of the SN location whereas its low-frequency accuracy order is O(κ⁸), where κ is the dimensionless wave number. For the diagonally lumped mass matrix (DLMM) constructed through the HRZ scheme, two optimal SN locations are identified, both away from third-points and of accuracy order O(κ⁸). That with the smallest error coefficient corresponds to the Lobatto 4- point integration rule. A special linear combination of CMM and DLMM with nodes at the Lobatto points yields an accuracy of O(κ¹0) without any increase in the computational effort over CMM. The effect of reduced integration (RI) on both mass and stiffness matrices is also studied. It is shown that singular mass matrices can be constructed with 2-and 3-point RI rules that display the same optimal accuracy of the exactly integrated case, at the cost of introducing spurious modes. The optimal SN location in two-dimensional, bicubic, isoparametric plane stress quadrilateral elements is briefly investigated by numerical experiments. The frequency accuracy of flexural modes is found to be fairly insensitive to that position, whereas for bar-like modes it agrees with the one-dimensional results.

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