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00 00 np N p2 n2 3 n2 6 4n2 10 9n2 15 16n2 Table 5.5.1: Shape function placement for Lagrange and hierarchical nite element approximations of degrees p = 1 2 3 4 on triangular elements with their number of parameters per element np and degrees of freedom N on a square with 2n2 elements. Circles
indicate additional shape functions located on a mesh entity.
of accuracy is a complex issue that depends on solution smoothness, geometry, and
the partial di erential system. We'll examine this topic in a later chapter. At least it
seems clear that bipolynomial bases are not competitive with hierarchical ones on square
elements. 5.5.1 Generation of Element Matrices and Vectors
The generation of the element sti ness and mass matrices and load vectors is largely
independent of the partial di erential system being solved however, let us focus on the
model problem of Section 3.1 in order to illustrate the procedures less abstractly. Thus,
1
consider the twodimensional Galerkin problem: determine u 2 HE satisfying A(v u) = (v f ) 1
8v 2 H0 (5.5.1a) 24 Mesh Generation and Assembly p
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Hierarchical
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Stencil
np N M n
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np N M n2
n2 00
n2
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0 1
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000 0
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00 000 0
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00 000 0
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00 000 0
11 111 1
00 000 0
11 111 1
00 000 0
11 111 1
00 000 0
11 111 1
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11 011 1
1
400 0 00 0
11 1 1
00 0 0
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11 1 1
00 1 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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11 1 1
00 0 0
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11 1 1 1
00 0 0 0
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11 1 1 1
00 0 0 0
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00 9 4n 16 9n2 2 1
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0 1
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0 1 11
0 00
1 11
0 00 1
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0 16n2 25 1 11
0 00
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0 00
1 11
0 00 3n2 12 5n2 17 8n2 11
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00 1
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0 Table 5.5.2: Shape function placement for bipolynomial Lagrange and hierarchical approximations of degrees p = 1 2 3 4 on square elements with their number of parameters
per element np and degrees of freedom N on a square with n2 elements. Circles indicate
additional shape functions...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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