A Hermite High-Order Finite Element in Structural Dynamics

The paper presents a review of a recently developedfinite-element technique based upon a three-dimensional Hermite interpolation for the evaluation of the natural modes of vibration of elastic structures, with spacial frequencies as high as possible, so as to make the technique useful for instance for structural acoustics applications. The finite-element unknowns are the nodal values of the unknown function (displacement), of its three first-order partial derivatives, of its three second-order mixed second derivatives, and of its third-order mixed derivative. The test-case used for the validation is the eigenproblem of the Laplacian, for a cubic domain, for which an exact solution is available. Applications include the evaluation of the natural frequencies of elastic plates, which are treated as three--dimensional objects, with only one element along the normal. The results obtained for simple geometries are highly encouraging, as they show that the method has an excellent rate of convergence, higher than that in standard finite-element methods. Thus, the extension to complex geometriesappears warranted. Unfortunately, existing geometry processor are not structured so as to provide the kind of data necessary for the type of element addressed here. Thus, a novel user-friendly formulation to generate complex three-dimensional geometries is presented.