Mass Matrix Templates: General Description and 1D Examples

  • Carlos A. FelippaEmail author
  • Qiong Guo
  • K. C. Park


This article is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The main focus is on adjusting the mass matrix. Two well known discretization methods, described in FEM textbooks since the late 1960s, lead to diagonally lumped and consistent mass matrices, respectively. Those models are sufficient to cover many engineering applications but for some problems 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. Templates have the virtue of producing a set of mass 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. Availability of free 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. The paper provides a general overview of the topic, and illustrates it with one-dimensional structural elements: bars and beams.



The first version of this paper was written during the 2004-2005 summer academic recesses while the first author was a visitor at CIMNE (Centro Internacional de Métodos Numéricos en Ingenieria) at Barcelona, Spain. The visits were partly supported by fellowships awarded by the Spanish Ministerio de Educación y Cultura during May-June of those years, and partly by the National Science Foundation under grant High-Fidelity Simulations for Heterogeneous Civil and Mechanical Systems, CMS-0219422. Thanks are due to Eugenio Oñate, director of CIMNE, for encouraging expanding the material to an expository paper. Thanks are also given to Manfred Bischoff and Anton Tkachuk for calling attention to recent work at the University of Stuttgart on mass modification methods, and providing articles in press.


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© CIMNE, Barcelona, Spain 2014

Authors and Affiliations

  1. 1.Department of Aerospace Engineering SciencesUniversity of Colorado at BoulderBoulderUSA
  2. 2.Shanghai Aircraft Design and Research InstituteCommercial Aircraft Corporation of China Ltd.ShanghaiChina
  3. 3.Department of Ocean EngineeringKAISTDaejonRepublic of Korea

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