A generalized infinite element is presented by combining following aspects: (1) The geometry mapping in the Cartesian system is chosen so as to facilitate the generation of infinite element mesh; (2) The decay variable is defined as the in situ major axis of the confocal ellipse where the field point is located, so it is dependent not only on the infinite-directional coordinate but on the finite-directional one(s); (3) The shape function is constructed so that it exactly satisfies the multipole expansion along the edges of the infinite element; (4) The conjugated weighting function is adopted to eliminate the harmonic terms from the integrands; and (5) the more proper form of phase factor and weighting factor is recommended. Compared with the Bettess element and the Astley element, the present element greatly reduces the element number within the finite element zone for problems with a large aspect ratio. Compared with the Burnett element and the modified Burnett element, the present element permits the free orientation of the infinite element, and thereby is suitable for more general exterior problems, either of a closed surface (source) or of an opening surface (source). Several typical examples are given and their results show that the generalized infinite element is robust and flexible.
An infinite element method is proposed to help solve practical problems in engineering and extend the applicability of infinite element. Based on the Helmholtz's equation, a novel governing equation is derived in terms of the modified sound pressure. The relative boundary conditions are established and the system matrices in using the combination of conventional finite element and new infinite element are subsequently formed. As a result, the use of coarser finite element meshes is permitted for a given frequency. The effectiveness and accuracy of this method are demonstrated in application to two typical examples.
SUMMARYIn this paper, the problems involved in the process of degeneration of quadrilateral element into triangular element are thoroughly analysed. The contents include the formulation of the geometry mapping induced by collapsing one side of the quadrilateral element and the construction of the shape functions. The study focuses ÿrst on a 4-node bilinear quadrilateral (Q4) element to 3-node constant strain triangular (CST ) element, and then on a 8-node serendipity (Q8) element to 6-node triangular element (T 6). In the analysis, the quadrilateral element and degenerate triangular element are assumed to be enclosed by straight edges. The theoretical results show that there is another better approach to realize the degeneration, and that even for conventional approach of degeneration we can give more reasonable explanation to the unclear problems like the CST property in degenerate CST element and the necessity of the additional terms in degenerate T 6 element.
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