User:Eml4500.f08.ateam.boggs.t/Lecture 39

Shape Functions


From Lecture 37-3,


 * $$ \tilde{d}_{6x1}^{(e)} = \tilde{T}_{6x6}^{(e)} \tilde{d}_{6x1}^{(e)}$$

Where $$ \tilde{d}^{(e)} $$ is known after solving the Finite Element System.

To compute $$ u(\tilde{x}), v(\tilde{x})$$,




 * $$ u(\tilde{x}) = U(\tilde{x}) \overrightarrow{ \tilde{i}} + V(\tilde{x}) \overrightarrow{ \tilde{j}} $$


 * $$ U(\tilde{x}) = U_x(\tilde{x}) \overrightarrow{ \tilde{i}} + U_y(\tilde{x}) \overrightarrow{ \tilde{j}} $$

To compute $$ U_x(\tilde{x}) $$ and $$ U_y(\tilde{x}) $$



\begin{bmatrix} U_x(\tilde{x})\\ U_y(\tilde{x}) \end{bmatrix} = R^{T} \begin{bmatrix} U(\tilde{x})\\ V(\tilde{x}) \end{bmatrix} $$

Where $$ U(\tilde{x}) $$ and $$ V(\tilde{x}) $$ are found



\begin{bmatrix} U(\tilde{x})\\ V(\tilde{x}) \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 & N_4 & 0 & 0 \\ 0 & N_2 & N_3 & 0 & N_5 & N_6 \end{bmatrix} \begin{bmatrix} \tilde{d_1}^{(e)}\\ \tilde{d_2}^{(e)}\\ \tilde{d_3}^{(e)}\\ \tilde{d_4}^{(e)}\\ \tilde{d_5}^{(e)}\\ \tilde{d_6}^{(e)} \end{bmatrix} $$



\mathbb{N} = \begin{bmatrix} N_1 & 0 & 0 & N_4 & 0 & 0 \\ 0 & N_2 & N_3 & 0 & N_5 & N_6 \end{bmatrix} $$

Therefore,



\begin{bmatrix} U_x(\tilde{x})\\ U_y(\tilde{x}) \end{bmatrix} = R^T \mathbb{N} \tilde{T}^{(e)} d^{(e)} $$