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Unformatted text preview: or the
secondorder 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 eightnoded 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 twodimensional 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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 Spring '14
 JosephE.Flaherty

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