User:Eml4500.f08.ramrod.c/12-1-08

Meeting 39 December 1, 2008


$$\mathbf{\tilde{d}^{(e)}}_{6x1} = \mathbf{\tilde{T}^{(e)}}_{6x6}\underbrace{\mathbf{d^{(e)}}_{6x1}}$$
 * $$\mbox{known after solving FE system}$$

Compute $$U(\tilde{x})\; \mbox{and} \; V(\tilde{x})$$


$$\mathbf{U(\tilde{x})} = U(\tilde{x})\vec{\tilde{i}} + V(\tilde{x})\vec{\tilde{j}}$$

$$\mathbf{U(\tilde{x})} = U_x(\tilde{x})\vec{i} + U_y({x})\vec{j}$$

Compute $$U(\tilde{x})\; \mbox{and}\; V(\tilde{x})$$ using Equations (1) and (2) from above

Compute $$U_x(\tilde{x}) \; \mbox{and} U_y(\tilde{x}) \; \mbox{from} \; U(\tilde{x}) \; \mbox{and} \; V(\tilde{x})$$
$$\begin{Bmatrix} U_x(\tilde{x})\\ V(\tilde{x}) \end{Bmatrix} = \mathbf{R}^T\begin{Bmatrix}U(\tilde{x})\\V(\tilde{x})\end{Bmatrix}$$


 * Note: The RT matrix can be found above.

$$\begin{Bmatrix}U(\tilde{x})\\V(\tilde{x})\end{Bmatrix} = \underbrace{\begin{bmatrix} N_1 & 0 & 0 & N_4 & 0 & 0\\ 0 & N_2 & N_3 & 0 & N_5 & N_6 \end{bmatrix}}_{\mathbb{N}(\tilde{x})}\begin{Bmatrix} \tilde{d}_1^{(e)}\\ \tilde{d}_2^{(e)}\\ \tilde{d}_3^{(e)}\\ \tilde{d}_4^{(e)}\\ \tilde{d}_5^{(e)}\\ \tilde{d}_6^{(e)} \end{Bmatrix}$$

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

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