Customizing high performance elements by fourier methods

Abstract

This paper illustrates, through a worked out example, the application of finite element templates to construct high performance bending elements for vibration and buckling problems. The example focuses on the improvement of the mass and geometric stiffness matrices of plane beams. Similar methods can be used for Kirchhoff plate bending elements, but this material is omitted because of space constraints. The process interweaves classical tools: Fourier analysis and orthogonal polynomials, with newer ones: finite element templates and computer algebra systems. Templates are parametrized algebraic forms that uniquely characterize an element population by a "genetic signature" defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design custom elements. For the beam example, Legendre polynomials are used to construct templates for the material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and stiffness that deliver matrices with desirable attributes for specific target situations. Three objectives are noted: high accuracy for vibration and buckling analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. This paper examines in some detail only the first objective.

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