Problem #13.8

$y^{(e)}(x) = [N(x)] vec{y}^(e) = lbrace N_i (x) N_j (x) rbrace delim{lbrace}{matrix{2}{1}{y_i y_j}}{rbrace}$ ,

$I = sum{e=1}{E=2}{I^(e)}$ : $I^(e) = {1/2}int{x_i}{x_j}{[-(dy/dx)^2 + 200x^2 y] dx}$ .

$I^(e) = {1/2} int{x_i}{x_j}{ lbrace -vec{y}^{(e)^T} [{dN(x)}/{dx}]^T [{dN(x)}/{dx}] vec{y}^(e) + 200 x^2 [N(x)] vec{y}^(e) rbrace }dx$

${partial I}/{partial y_m} = sum{e=1}{E}{{partial I^(e)}/{partial y_m}} = 0; m=1,2,P. P=3; E=2.$

$sum{e=1}{E}{{partial I^(e)}/{partial vec{y}^(e)}} = sum{e=1}{E}{int{x_i}{x_j}{lbrace -[{dN(x)}/{dx}]^T [{dN(x)}/{dx}] vec{y}^(e) + 100 x^2 [N(x)]^T rbrace}} = vec{0}$

$sum{e=1}{E}{[S^(e)]vec{y}^(e)} = sum{e=1}{E}{vec{Q}^(e)}$ : $int{x_i}{x_j}{lbrace [{dN(x)}/{dx}]^T [{dN(x)}/{dx}] vec{y}^(e) rbrace dx}$

$vec{Q}^(e) = int{x_i}{x_j}{100 x^2 [N(x)]^T dx}$

$[N(x)] = [({x_j - x}/{l_e})({x-x_i}/{l_e})]$ , $[{dN(x)}/{dx}] = [({-1/{l_e}})(1/{l_e})]$ .

$l_1=l_2={1/2}; x_i=0 x_j=0.5 for e=1; x_i=0.5 x_j=1 for e=2;$

$[S^(e)] = int{x_i}{x_j}{[delim{lbrace}{matrix{2}{1}{{-1/{l_e}} {1/l_e}}}{rbrace} delim{lbrace}{matrix{1}{2}{{-1/l_e} {1/l_e}}}{rbrace} ]}dx =$ ${1/l_e}delim{[}{matrix{2}{2}{1 {-1} {-1} 1}}{]}$ delim{[}{matrix{2}{2}{2 {-2} {-2} 2}}{]} ; e=1,2;

$vec{Q}^(e) = 100 int{x_i}{x_j}{ lbrace matrix{2}{1}{({x_j - x}/{l_e})({x-x_i}/{l_e})} rbrace x^2} dx=$ ${100/l_e} int{x_i}{x_j}{lbrace matrix{2}{1}{{x_j x^2 - x^3}{x^3 - x_i x^2}} rbrace} dx=$ ${100/l_e}[ lbrace matrix{2}{1}{{{x_j/3} x^3 - {1/4}x^4}{{1/4}x^4 - {1/3}x_i x^3}} rbrace ]delim{|}{matrix{2}{1}{x_j x_i}}{}=$ ${100/l_e}[ lbrace matrix{2}{1}{{{{1/3}{x_j}^4} - {1/4}{x_j}^4}{{1/4}{x_j}^4 - {1/3}x_i {x_j}^3}} rbrace -lbrace matrix{2}{1}{{ {x_j/3} {x_i}^3 - {1/4}{x_i}^4}{{1/4}{x_i}^4 - {1/3}{x_i}^4}} rbrace ]=$ .

$vec{Q}^(1)=$

CE 5143 Homework 6 Due 11/15/07

Problem #13.8

Problem #13.11

Problem #13.13

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