TY - EJOU
AU - Felippa, Carlos A.
TI - Mass-Stiffness Templates for Cubic Structural Elements
T2 - Computer Modeling in Engineering \& Sciences
PY - 2021
VL - 129
IS - 3
SN - 1526-1506
AB - 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(κ^{8}), 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(κ^{8}). 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(κ^{1}0) 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.
KW - Structural dynamics; Lagrangian elements; finite elements; cubic shape functions; bar; plane stress; mass; stiffness; vibration; wave propagation; Fourier analysis; dispersion; templates
DO - 10.32604/cmes.2021.016803