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Unformatted text preview: or the second-order hierarchical shape functions of Section 4.4. Let the four vertex nodes be numbered (1 1), (2 1), (1 1), and (2 1) and the four midside nodes be numbered (3 1), (1 3), (2 3), and (3 2). Use the serendipity shape functions of Problem 4.3.1 to map the canonical 2 2 square element onto an eight-noded quadrilateral element with curved sides in the (x y)-plane. Assume that the vertex and midside nodes of the physical element have the same numbering as the canonical element but have coordinates at (xij yij ), i j = 1 2 3, i = j 6= 3. Can the Jacobian of the transformation vanish for some particular choices of (x y)? (This is not a simple question. It su ces to give some qualitative reasoning as to how and why the Jacobian may or may not vanish.) 22 Mesh Generation and Assembly 2. Consider the transformation (5.4.7) of Example 5.4.3 with x5 = y5 = 1=4 and sketch the element in the (x y)-plane. Sketch the element for some choice of x5 = y5 < 1=4. 5.5 Generation of Element Matrices and Vectors and Their Assembly Having discretized the domain, the next step is to select a nite element basis and generate and assemble the element sti ness and mass matrices and load vectors. As a review, we summarize some of the two-dimensional shape functions that were developed in Chapter 4 in Tables 5.5.1 and 5.5.2. Nodes are shown on the mesh entities for the Lagrangian and hierarchical shape functions. As noted in Section 5.3, however, the shape functions may be associated with the entities without introducing modal points. The number of parameters np for an element having order p shape functions is presented for p = 1 2 3 4. We also list an estimate of the number of unknowns (degrees of freedom) N for scalar problems solved on unit square domains using uniform meshes of 2n2 triangular or n2 square elements. Both the Lagrange and hierarchical bases of order p have the same number of parameters and degrees of freedom on the uniform triangular meshes. Without constraints for Dirichlet data, the n...
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