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- I wish to set up the node equations for a 2D heated plate with boundary conditions.

I wish to set up the node equations for a 2D heated plate with boundary conditions. I understand how to do this in 1D but have not found a suitable example problem worked out in 2D and examples I have seen are very involved and complex. @pasmith showed me you to set up the 1D problem as follows;

##N_i## are test functions with [itex]T[/itex] approximated by ##T_1N_1(x) + T_2N_2(x)##. If so, then ##\frac{d^2T}{dx^2} + kT(x) = -f(x)##where ##f(x)## is a known source term gives ;

$$\begin{multline}

\int_{x_1}^{x_2} N_i(x)\frac{d^2T}{dx^2} + kN_i(x)T(x) + N_i(x)f(x)\,dx \\=

\left[ N_i(x)\frac{dT}{dx}\right]_{x_1}^{x_2}

+\int_{x_1}^{x_2} -\frac{dN_i}{dx}\frac{dT}{dx} + kN_i(x)T(x) + N_i(x)f(x)\,dx

\end{multline}$$which results in ;

$$\begin{multline}

\begin{pmatrix}

\int_{x_1}^{x_2} \left(\frac{dN_1}{dx}\right)^2\,dx & \int_{x_1}^{x_2} \frac{dN_1}{dx}\frac{dN_2}{dx} \,dx \\

\int_{x_1}^{x_2} \frac{dN_1}{dx}\frac{dN_2}{dx} \,dx & \int_{x_1}^{x_2} \left(\frac{dN_2}{dx}\right)^2\,dx

\end{pmatrix}

\begin{pmatrix} T_1 \\ T_2 \end{pmatrix} \\

- k\begin{pmatrix} \int_{x_1}^{x_2} N_1^2(x)\,dx & \int_{x_1}^{x_2} N_1(x)N_2(x)\,dx \\

\int_{x_1}^{x_2} N_1(x)N_2(x)\,dx & \int_{x_1}^{x_2} N_2^2(x)\,dx \end{pmatrix}

\begin{pmatrix} T_1 \\ T_2 \end{pmatrix} \\

=

\begin{pmatrix}

N_1(x_2) \frac{dT}{dx}(x_2) - N_1(x_1) \frac{dT}{dx}(x_1) \\

N_2(x_2) \frac{dT}{dx}(x_2) - N_2(x_1) \frac{dT}{dx}(x_1) \end{pmatrix} +

\begin{pmatrix}

\int_{x_1}^{x_2} N_1(x)f(x)\,dx \\

\int_{x_1}^{x_2} N_2(x)f(x)\,dx

\end{pmatrix}.\end{multline}$$

I wish to amend this for a 2D problem. My book starts with an element equation ##u(x, y) = a_0 + a_{1,1}x + a_{1,2} y## and test functions like this ##N_1 = 1/ 2Ae [(x_2 y_3 − x_3 y_2) + (y_2 − y_3)x + (x_3 − x_2)y]## but I can't seem to put things together.

##N_i## are test functions with [itex]T[/itex] approximated by ##T_1N_1(x) + T_2N_2(x)##. If so, then ##\frac{d^2T}{dx^2} + kT(x) = -f(x)##where ##f(x)## is a known source term gives ;

$$\begin{multline}

\int_{x_1}^{x_2} N_i(x)\frac{d^2T}{dx^2} + kN_i(x)T(x) + N_i(x)f(x)\,dx \\=

\left[ N_i(x)\frac{dT}{dx}\right]_{x_1}^{x_2}

+\int_{x_1}^{x_2} -\frac{dN_i}{dx}\frac{dT}{dx} + kN_i(x)T(x) + N_i(x)f(x)\,dx

\end{multline}$$which results in ;

$$\begin{multline}

\begin{pmatrix}

\int_{x_1}^{x_2} \left(\frac{dN_1}{dx}\right)^2\,dx & \int_{x_1}^{x_2} \frac{dN_1}{dx}\frac{dN_2}{dx} \,dx \\

\int_{x_1}^{x_2} \frac{dN_1}{dx}\frac{dN_2}{dx} \,dx & \int_{x_1}^{x_2} \left(\frac{dN_2}{dx}\right)^2\,dx

\end{pmatrix}

\begin{pmatrix} T_1 \\ T_2 \end{pmatrix} \\

- k\begin{pmatrix} \int_{x_1}^{x_2} N_1^2(x)\,dx & \int_{x_1}^{x_2} N_1(x)N_2(x)\,dx \\

\int_{x_1}^{x_2} N_1(x)N_2(x)\,dx & \int_{x_1}^{x_2} N_2^2(x)\,dx \end{pmatrix}

\begin{pmatrix} T_1 \\ T_2 \end{pmatrix} \\

=

\begin{pmatrix}

N_1(x_2) \frac{dT}{dx}(x_2) - N_1(x_1) \frac{dT}{dx}(x_1) \\

N_2(x_2) \frac{dT}{dx}(x_2) - N_2(x_1) \frac{dT}{dx}(x_1) \end{pmatrix} +

\begin{pmatrix}

\int_{x_1}^{x_2} N_1(x)f(x)\,dx \\

\int_{x_1}^{x_2} N_2(x)f(x)\,dx

\end{pmatrix}.\end{multline}$$

I wish to amend this for a 2D problem. My book starts with an element equation ##u(x, y) = a_0 + a_{1,1}x + a_{1,2} y## and test functions like this ##N_1 = 1/ 2Ae [(x_2 y_3 − x_3 y_2) + (y_2 − y_3)x + (x_3 − x_2)y]## but I can't seem to put things together.

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