兰器装 二 留- ★》--之Nm D if F-NI D h2-Dcme -∑V%,,-∑N5 If we consider rectangular elements as in Figure c (an intentionally crude mesh for illustrative purposes), the problems are apparent. Unless the elements are very small, the area of the domain excluded from the model (the shaded area in the figure) may be significant. For the case depicted, a large number of very small square elements best approximates the geometry. At this point, you may think, Why not use triangular and rectangular elements in the same mesh to improve the model? Indeed, a combination of the element types can be used to improve the geometric accuracy of the model. The shaded areas of Figure c could be modeled by three-node triangular elements. Such combination of element types may not be the best in terms of solution accuracy since the rectangular element and the triangular element have, by necessity, different order polynomial representations of the field variable. The field variable is continuous across such element boundaries; this is guaranteed by the finite element formulation. However, conditions on derivatives of the field variable for the two element types are quite different. On a curved boundary such as that shown, the triangular element used to fill the “gaps” left by the rectangular elements may also have adverse aspect ratio characteristics. Now examine Figure d, which shows the same area meshed with rectangular elements and a new element applied near the periphery of the domain. The new element has four nodes, straight sides, but is not rectangular. (Please note that the mesh shown is intentionally coarse for purposes of illustration.) The new element is known as a general two-dimensional quadrilateral element and is seen to mesh ideally with the rectangular element as well as approximate the curved boundary, just like the triangular element. The four-node quadrilateral element is derived from the four-node rectangular element (known as the parent element) element via a mapping process. Figure above shows the parent element and its natural (r, s) coordinates and the quadrilateral element in a global Cartesian coordinate system. The geometry of the quadrilateral element is described by Mapping of a parent element into an isoparametric element. where the Gi (x , y) can be considered as geometric interpolation functions, and each such function is associated with a particular node of the quadrilateral element. Given the geometry and the form of Equations above, each function Gi (x , y) must evaluate to unity at its associated node and to zero at each of the other three nodes. These conditions are exactly the same as those imposed on the interpolation functions of the parent element. Consequently, the interpolation functions for the parent element can be used for the geometric functions, if we map the coordinates so that Consequently, the geometric expressions become Clearly, we can also express the field variable variation in the quadrilateral element as Since the same interpolation functions are used for both the field variable and description of element geometry, the procedure is known as isoparametric (constant parameter) mapping. The element defined by such a procedure is known as an isoparametric element