Construction of Customized Mass-Stiffness Pairs Using Templates

Abstract

This article is a tutorial exposition ofthe template approach to the construction ofcustomized mass- stiffness pairs for selected applications in structural dynamics. The exposition focuses on adjusting the mass matrix while a separately provided stiffness matrix is kept fixed. Two well known kinetic- energy discretization methods described in FEM textbooks since the mid 1960s lead to diagonally- lumped and consistent mass matrices, respectively. These two models are sufficient to cover many engineering applications. Occasionally, however, they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. This approach is discussed in this paper using one-dimensional structural elements as examples. Templates have the virtue ofproducing a set ofmass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance and momentum conservation. In particular, the diagonally-lumped and consistent versions can be obtained as instances. Thus those standard models are not excluded. Availability offree parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes.