Customizing the mass and geometric stiffness of plane beam elements by Fourier methods

Abstract

Teaches by example the application of finite element templates in constructing high performance elements. The example discusses the improvement of the mass and geometric stiffness matrices of a Bernoulli-Euler plane beam. This process interweaves classical techniques (Fourier analysis and weighted orthogonal polynomials) with newer tools (finite element templates and computer algebra systems). Templates are parameterized 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 this example weighted orthogonal polynomials are used to construct templates for the beam material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and geometric stiffness that deliver matrices with desirable properties when used in conjunction with the well-known Hermitian beam material stiffness. For mass-stiffness combinations, three objectives are noted: high accuracy for vibration analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. Only the first objective is examined in detail.

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