User:Eml5526.s11.team01.roark/Mtg19

[[media: Fe1.s11.mtg19.djvu| Mtg 19]]: Mon, 14 Feb 11 [[media: Fe1.s11.mtg19.djvu| Page 19-1]] Discrete WF (DWF): [[media: Fe1.s11.mtg18.djvu| p. 18-2]] continued
 * \mathbf, \boldsymbol
 * Orthogonal Basis functions ([[media: Fe1.s11.mtg4.djvu|(0) p. 10-3]]) do not equal orthogonal matrices
 * [[media: Fe1.s11.mtg4.djvu|(3) & (5) p. 18-1]]: $$\displaystyle ~cos\phi $$ and $$\displaystyle cos\frac{\pi }{4}$$ play important role of the constant in a family of basis functions

Approximate [[media: Fe1.s11.mtg10.djvu|(1) p. 10-3]] ($$\displaystyle {{w}^{h}}$$) and [[media: Fe1.s11.mtg10.djvu|(2) p. 4-2]] ($$\displaystyle {{u}^{h}}$$, Approximate for “trial solution” u) [[media: Fe1.s11.mtg16.djvu|(3) p. 16-2]]:
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$$\displaystyle \tilde{m}\left( {{w}^{h}},{{u}^{h(s)}} \right)=\underset{i}{\mathop \sum }\,\underset{j}{\mathop \sum }\,{{c}_{i}}\underbrace{\tilde{m}\left( {{b}_{i}},{{b}_{j}} \right)}_d_{j}^{s}$$
 *  (1)
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$$\displaystyle {{\tilde{M}}_{ij}}=\underset{\alpha }{\overset{\beta }{\mathop \int }}\,{{b}_{i}}\tilde{m}{{b}_{j}}dx$$ [[media: Fe1.s11.mtg19.djvu| Page 19-2]] [[media: Fe1.s11.mtg19.djvu|(1) p. 16-2]]
 *  (2)
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$$\displaystyle \tilde{f}\left( {{w}^{h}} \right)=\sum\limits_{i}\underbrace{\tilde{f}\left( {{b}_{i}} \right)}_$$
 *  (1)
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$$\displaystyle {{\tilde{F}}_{i}}={{b}_{i}}\underbrace{(\alpha )}_{\begin{smallmatrix} {{\Gamma }_{h}} \\ (nat'l\,\,\,b.c.) \end{smallmatrix}}h+\int_{\alpha }^{\beta }{{{b}_{i}}fdx}$$ Note:
 *  (2)
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[[media: Fe1.s11.mtg19.djvu|(1) p. 19-1]]

[[media: Fe1.s11.mtg10.djvu| p. 10-3]]
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$$\displaystyle {{u}^{h}}(x,t)=\sum\limits_{j}{{{d}_{j}}(t){{b}_{j}}(x)}$$
 *  (3)
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$$\displaystyle {{u}^{h}}^{(s)}(x,t)=\frac{\partial {{t}^{s}}}{{u}^{h}}(x,t)=\sum\limits_{j}{\underbrace{{{d}_{j}}^{s}(t)}_{\frac{\partial {{t}^{s}}}{{d}_{j}}(t)}{{b}_{j}}(x)}$$ End Note Note:
 *  (4)
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Homogenous essential boundary condition ([[media: Fe1.s11.mtg16.djvu| p. 16-4]]) [[media: Fe1.s11.mtg16.djvu|(3) p. 16-1]]
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$$\displaystyle w\left( \beta \right)\cong {{w}^{h}}\left( \beta  \right)=0$$
 *  (5)
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$$\displaystyle \sum\limits_{i}{{{c}_{i}}{{b}_{i}}\left( \beta \right)=0}$$ End Note
 *  (6)
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[[media: Fe1.s11.mtg19.djvu| Page 19-3]] DWF
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Find $$\displaystyle \left\{ {{d}_{j}} \right\}$$ st
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$$\displaystyle {{u}^{h}}(\beta )=\sum\limits_{j}{{{d}_{j}}\left( t \right){{b}_{j}}\left( \beta \right)}=g$$ And
 *  (1)
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$$\displaystyle \sum\limits_{i}{{{c}_{i}}\left[ \sum\limits_{j}{\left\{ {{{\tilde{M}}}_{ij}}{{d}_{j}}^{s}+{{{\tilde{K}}}_{ij}}{{d}_{j}} \right\}}-{{{\tilde{F}}}_{i}} \right]}=0$$ $$\displaystyle \forall {{c}_{i}}$$ st
 *  (2)
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$$\displaystyle \underbrace{\sum\limits_{i}{{{c}_{i}}{{b}_{i}}\left( \beta \right)}=0}_{Constr.\,on\,\left\{ {{c}_{i}} \right\}}$$
 *  (3)
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The above is also called DWF-C (constrained)