#### Date of Award

Spring 1-1-2012

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Aerospace Engineering Sciences

#### First Advisor

Carlos A. Felippa

#### Second Advisor

H. Jerry Qi

#### Abstract

This thesis studies the construction of improved mass matrices for dynamic structural analysis using the finite element method (FEM) for spatial discretization. Two kinetic-energy discretization methods described in FEM textbooks since the mid-1960s lead to diagonally-lumped and consistent mass matrices, respectively. While these well-known models are sufficient to cover many engineering applications, they may fail to satisfy customized optimality conditions, such as delivering better accuracy in the low frequency (long wavelength) limit, which is important in structural dynamics and vibrations. Such gaps can be filled with a more general approach that relies on the use of mass templates. These are algebraic forms that carry free parameters. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, observer invariance and linear momentum conservation. In particular, the diagonally-lumped and consistent mass versions can be obtained as instances; thus those standard models are not excluded. The presence of free parameters, however, allows the mass matrix to be customized to specific needs. A mass template is called optimal if it meets a quantifiable "best" criteria, such as highest low-frequency accuracy, for certain values of the parameters. The present work develops such conditions by studying the propagation of two types of plane waves: P (pressure) and S (shear), over regular, infinite, square-cell FEM lattices of isotropic plates. Such studies are equivalent to directional Fourier analysis. Only one-parameter templates, obtained by linear weighting lumped and quasi-consistent mass matrix instances, are considered. Using a computer algebra system (CAS), exact dispersion expressions are obtained for the two elements under study. In addition to the free parameter, dispersion is found to depend on three factors: Poisson's ratio, propagation angle with respect to lattice principal directions, and wave type (P or S). Exact expressions are Taylor expanded in the low frequency limit and matched, using the template parameter, with the continuum dispersion up to maximum possible order in the wavenumber. Matches are further averaged over propagation angle and Poisson's ratio ranges to provide recommended values for use in existing FEM codes. The present work represents the first work of this nature for two-dimensional finite elements. It was made possible by steady improvements in CAS software, as well as CPU and RAM computer resources. As summarized in the Conclusions Chapter, these initial results suggest future-work extensions that remove several of the simplifying assumptions made in this study.

#### Recommended Citation

Guo, Qiong, "Developing optimal mass matrices for membrane triangles with corner drilling freedoms" (2012). *Aerospace Engineering Sciences Graduate Theses & Dissertations*. 62.

https://scholar.colorado.edu/asen_gradetds/62