Coons Patch, Hermite Interpolation, and High-Order Finite Elements in Structural Dynamics

The paper presents a novel finite element for the evaluation of the natural modes of vibrations of complex structures. The element is based upon a three-dimensional extension of the Coons patch technique, combined with the fact that the generating lines are obtained using the Hermite interpolation technique; the resulting finite-element unknowns are the nodal values of: (i) the unknownfunction (the displacement vector in our case), and (ii) the Cartesian components of its gradient. In addition, the paper presents a review of recent work by the authors on another closely related element, which is an extension to complex configurations of the Hermite element, which in turn is based upon the three-dimensional extension of the Hermite interpolation; in this case, the finite-elementunknowns are the nodal values of: (i) the unknown function, (ii) the Cartesian components of its gradient, (iii) its three second?order mixed derivatives, and (iv) its third-order mixed derivative. The objective of these methods is the user?friendly evaluation of natural modes of vibration of elastic structures, as used in multi-disciplinary optimization; accordingly, in order to validate and assess the two methods, numerical results for simple test cases are included; we concentrate on the natural frequencies of elastic plates (which are treated as three-dimensional structures, with only one elementalong the normal). In addition, in view of the fact that the ultimate objective is the applicability of the techniques for arbitrary geometries, results for a relatively complex structure (that is, a simplifiedmodel of a wing box) are included.