A review of 4-node, 12 degrees-of-freedom quadrilateral elements for thin plates is presented. A new element called DKQ is discussed. The formulation is based on a generalization of the efficient and reliable triangular element DKT presented in References 1 and 2 and on the rectangular element QC presented in Reference 3. These elements are derived using the so-called discrete Krichhoff technique. A detailed numerical evaluation of the behaviour of the DKQ element for the computation of displacements and stresses for thin plate bending problems is presented and discussed. The DKQ element appears to be a simple and reliable 12 degrees-of-freedom thin plate bending element.

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A nonstandard wave equation, established by Galbrun in 1931, is used to study sound propagation in nonuniform flows. Galbrun's equation describes exactly the same physical phenomenon as the linearized Euler's equations (LEE) but is derived from an Eulerian-Lagrangian description and written only in term of the Lagrangian perturbation of the displacement. This equation has interesting properties and may be a good alternative to the LEE: only acoustic displacement is involved (even in nonhomentropic cases), it provides exact expressions of acoustic intensity and energy, and boundary conditions are easily expressed because acoustic displacement whose normal component is continuous appears explicitly. In this paper, Galbrun's equation is solved using a finite element method in the axisymmetric case. With standard finite elements, the direct displacement-based variational formulation gives some corrupted results. Instead, a mixed finite element satisfying the inf-sup condition is proposed to avoid this problem. A first set of results is compared with semianalytical solutions for a straight duct containing a sheared flow (obtained from Pridmore-Brown's equation). A second set of results concerns a more complex duct geometry with a potential flow and is compared to results obtained from a multiple-scale method (which is an adaptation for the incompressible case of Rienstra's recent work).