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Unformatted text preview: ted with the vertices, the
next three are quadratic corrections at the midsides, the next three are cubic corrections
at the midsides, and the last is a cubic \bubble function" associated with the centroid
An array implementation, as described by (5.5.6) and Examples 5.5. 1 and 5.5.2,
may be the simplest data structure however, implementations with structures linked to
geometric entities (Section 5.3) are also possible.
Substituting the polynomial representation (5.5.6) into the transformed strain energy
expression (5.5.4b) and external load (5.5.4a) yields
Ae(V U ) = dT (Ke + Me)ce
e (5.5.7a) 28 Mesh Generation and Assembly
6, 9 0
0 1 4, 7 1
0 ξ 2 Figure 5.5.2: Shape function placement and numbering for a hierarchical cubic approximation on a canonical right 45 triangle.
(V f )e = dT fe
where K= ZZ e (5.5.7b) g1eN NT + g2e(N NT + N NT ) + g3eN NT ] det(Je)d d (5.5.8a) 0 M= ZZ e qNNT det(Je)d d (5.5.8b) Nf det(J )d d : (5.5.8c) 0 f= ZZ e e 0 Here, Ke and Me are the element sti ness and mass matrices and fe is the element load
vector. Numerical integration will generally be necessary to evaluate these arrays when
the coordinate transformation is not linear and we will study procedures to do this in
Element mass and sti ness matrices and load vectors are generated for all elements
in the mesh and assembled into their proper locations in the global sti ness and mass
matrix and load vector. The positions of the elemental matrices and vectors in their
global counterparts are determined by their indexing. In order to illustrate this point,
consider a linear shape function on an element with Vertices 4, 7, and 8 as shown in
Figure 5.5.3. These vertex indices are mapped onto local indices, e.g., 1, 2, 3, of the
canonical element and the correspondence is recorded as shown in Figure 5.5.3. After
generating the element matrices and vectors, the global indexing determines where to add 5.5. Element Matrices and Their Assembly 29 these entries into the global sti nes and mass matrix and load v...
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- Spring '14