User:Egm5526.s11.team2.sandhu/newpage

new page home work 5 =Problem Weak form solution using LLEIF= Solve the following discrete weak form with Linear Lagrange Element Interpolation Basis.


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$$  \displaystyle \sum {{c_i}\left[ {\sum {{{\tilde K}_{ij}}{d_j} - {{\tilde F}_i}} } \right]} = 0 $$     (3.7.1)
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Where


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$$  \displaystyle {{\tilde K}_{ij}} = \int\limits_0^1 {{{b'}_i}(2 + 3x){{b'}_j}dx} $$     (3.7.2)
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$$  \displaystyle {F_i} = {b_i}\left( 1 \right)\left( {12} \right) + \int_0^1 {{b_i}\left( x \right)5x} dx $$ (3.7.3)
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Such that


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$$  \displaystyle {u^h}\left( 0 \right) = \sum {{d_j}{b_j}\left( 0 \right) = 4} $$     (3.7.4)
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$$  \displaystyle {w^h}\left( 0 \right) = \sum {{c_i}{b_i}\left( 0 \right) = 0} $$     (3.7.5)
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d. Plot $${u^h}$$ vs $${u}$$
Plot $${u^h}\left( {0.5} \right) - u\left( {0.5} \right)$$ vs n (total no. of dof's)

a. Using LLEBF in constraint breaking solution(CBS)
We have following constraint to be satisfied to use weak form


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$$  \displaystyle {w^h}\left( 0 \right) = \sum {{c_i}{b_i}\left( 0 \right) = 0} $$     (3.7.6)
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At node 1 (i.e. at x=0) all the basis function are zero except 1.


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$$  \displaystyle {w^h} = {c_1}{b_1} + {c_2}{b_2} + {c_3}{b_3} + {c_4}{b_4} + \cdots $$     (3.7.7)
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$$  \displaystyle {w^h} = {c_1}(1) + {c_2}(0) + {c_3}(0) + {c_4}(0) + \cdots $$     (3.7.8)
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$$  \displaystyle {w^h} = {c_1} $$     (3.7.9)
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For
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$$  \displaystyle {w^h}\left( 0 \right) = 0 $$     (3.7.10)
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$$  \displaystyle {c_1} = 0 $$     (3.7.11)
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To satisfy homogeneous boundary condition while using Linear Lagrange Element Interpolation Functions, we just have to select our $$w^h$$ such that $${c_1} = 0$$.

c Formulation of K and F on element level
We can divide the total domain into elemental domain.


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$$  \displaystyle \Omega = \bigcup\limits_{e = 1}^{nel} $$     (3.7.12)
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Values of K and F will be summation of K and F over the elements
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$$  \displaystyle {{\tilde K}_{ij}} = \sum\limits_{e = 1}^{nel} {K_{ij}^e} $$     (3.7.13)
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$$  \displaystyle K_{ij}^e = \int\limits_ {{{b'}^e}_i(2 + 3x){{b'}^e}_jdx} $$     (3.7.14)
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$$  \displaystyle {{\tilde F}_i} = \sum\limits_{e = 1}^{nel} {F_i^e} $$     (3.7.15)
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$$  \displaystyle {F_i}^e = \int_ {b_i^e5xdx} $$     (3.7.16)
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There will be two basis function for each element. Generalised values of these basis function will be as follow


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$$  \displaystyle {b_e} = \frac $$     (3.7.17)
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$$  \displaystyle {b_{e + 1}} = \frac $$     (3.7.18)
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d. Solution using Matlab
Plot of exact solution and approximate solution with 2 elements



Plot of exact solution and approximate solution with 4 elements



Plot of exact solution and approximate solution with 6 elements



Plot of exact solution and approximate solution with 8 elements



Plot of error vs dof's


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$$  \displaystyle err = u\left( {0.5} \right) - {u^h}\left( {0.5} \right) $$     (3.7.19)
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$$  \displaystyle dof's = nel + 1 $$     (3.7.20)
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where nel= number of elements.



for 16 elements

dof's=16+1=17



comment
Solution is converging with increasing no. of elements. For n=16 error is less than 0.0001.