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

Teaches by example the application of finite element templates in construct ing high performance elements. The example discusses the improvement of the mass and geometric stiffness matrices of a Bernoulli-Euler beam. This proc ess interweaves classical techniques (Fourier analysis and weighted orthogo nal polynomials) with newer tools (finite element templates and computer al gebra systems). Templates are parameterized algebraic forms that uniquely c haracterize an element population by a "genetic signature" defined by the s et of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design "cus tom" elements. For this example weighted orthogonal polynomials are used to construct templates for the beam material stiffness, geometric stiffness a nd mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and geometric stiffness that deli ver matrices with desirable properties when used in conjunction with well-k nown Hermitian beam material stiffness. For mass-stiffness combinations, th ree objectives are noted: high accuracy for vibration analysis, wide separa tion of acoustic and optional branches, and low sensitivity to mesh distort ion and boundary conditions. Only the first objective is examined in detail .