This is the second part of a two part paper on three‐dimensional finite elements with rotational degrees of freedom (DOF). Part II introduces a solid tetrahedron element having 3 translational and 3 rotational DOF per node. The corner rotations are introduced by transformation of the midside translational DOF of a 10‐node tetrahedron element. To further enhance the element performance a least squares strain extraction technique is also implemented to develop the stiffness matrix with a desired field. The strain smoothing improves performance without causing a loss in generality. As with the hexahedron in Part I, the element stiffness is augmented with a small penalty stiffness to eliminate any possible spurious zero energy modes. The new tetrahedron element passes the patch test and demonstrates much improved performance over the 4‐node translational DOF only (constant strain) tetrahedron element.

SUMMARYThis is the first of a two part paper on three-dimensional finite elements with rotational degrees of freedom (DOF). Part I introduces an 8-node solid hexahedron element having three translational and three rotational DOF per node. The corner rotations are introduced by transformation of the midside translational DOF of a 20-node hexahedron element. The new element produces a much smaller effective band width of the global system equations than does the 20-node hexahedron element having midside nodes.A small penalty stiffness is introduced to augment the usual element stiffness so that no spurious zero energy modes are present. The new element passes the patch test and demonstrates greatly improved performance over elements of identical shape but having only translational DOF at the corner nodes.

This paper presents an efficient numerical integration scheme for evaluating the matrices (stiffness, mass, stress-stiffness and thermal load) for a doubly curved, multilayered, composite, quadrilateral shell finite element. The element formulation is based on three-dimensional continuum mechanics theory and it is applicable to the analysis of thin and moderately thick composite shells.The conventional formulation requires a 2 x 2~2 or 2 x 2~ 1 Gauss integration per layer for the calculation of element matrices. This method becomes uneconomical when a large number of layers is used owing to an exccssive amount of computations. The present formulation is based on explicit separation of the thickness variable from the shell surface parallel variables. With the through-thickness variables separated, they are combined with the thickness dependent material properties and integrated separately. The element matrices are computed using the integrated material matrices and only a 2 x 2 spatial Gauss integration scheme. The response results using the present formulation are identical to those obtained using the conventional formulation. For a small number of layers, the present method requires slightly more CPU time. However, for a larger number of layers, numerical data are presented to demonstrate that the present forinulation is an order-of-magnitude economical compared to the conventional scheme.

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SUMMARYAcoustic algebraic eigenvalue analysis by the Boundary Element Method (BEM) can be formulated by the Dual Reciprocity Method (DRM) of Nardini and Brebbia or by the Complementary Function-Particular Integral Method (PIM) proposed by Ahmad and Banerjee. But both DRM and PIM require inversion of a matrix of size at least as large as the system matrices before the equations can be cast in the form of generalized eigensystem. This makes these methods inefficient for large problems of practical interest. In this paper, a rather simple technique is proposed which eliminates the need to invert any matrix in the process of setting up the algebraic eigenvalue problem, especially for the most important case where all the boundary walls are acoustically hard (aP/an = 0). A few example problems having known analytical and experimental results are solved in order to demonstrate the validity of the new technique. It is also demonstrated that, unlike in elasticity, here the boundary element domain must be adequately zoned or an adequate number of internal points must be incorporated in order to solve truly 2-D or 3-D problems.