Resolution of the 1D Poisson's equation using local Shape Functions

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The problem to solve is:

$A(\varphi) = \frac{d}{dx} \left( k \frac{d \varphi}{dx} \right) + Q = 0 ~~ in ~ 0 \le x \le l$

$B(\varphi) = \begin{cases} \varphi - \overline{\varphi} = 0 & in ~ x = 0 \\ k \frac{d \varphi}{dx} + \overline{q} = 0 & in ~ x = l \end{cases}$

with the following analytical solution:

$\varphi (x) = - \frac{Q}{2 k} x^2 + \frac{- \overline q + Q l}{k} x + \overline \varphi$

$k \nabla \varphi (x) = Q (l - x) - \overline q$

The weak form of the integral equation is:

${ \int_0^l \frac{d W_i}{dx} k \sum_{j=1}^n \frac{d N_j}{dx} a_j dx = \int_0^l W_i Q dx + \left[ W_i q \right ]_{x=0} - \left[ W_i \overline q \right ]_{x=l} }$

and the approach solution is written by using the shape functions as:

$\varphi (x) \cong \hat \varphi (x) = \sum_{i=1}^n N_i (x) \varphi_i$

Resolution using 1 element with two nodes

One single element means to cover the entire domain with the local functions.

$\hat \varphi (x) = N_1 (x) \varphi_1 + N_2 (x) \varphi_2$

$\frac{ d \hat \varphi (x)}{dx} = \frac{ d N_1 (x) }{dx} \varphi_1 + \frac{ d N_2 (x) }{dx} \varphi_2$

By using the Galerkin method ( $W_i = N_i \,$ , it can be obtained:

$\int_0^l \frac{d N_i}{dx} k \left ( \frac{d N_1}{dx} \varphi_1 + \frac{d N_2}{dx} \varphi_2 \right ) dx = \int_0^l N_i Q dx + \left[ N_i q \right ]_{x=0} - \left[ N_i \overline q \right ]_{x=l} \qquad i=1,2$

for i=1:

$\int_0^l \frac{d N_1}{dx} k \left ( \frac{d N_1}{dx} \varphi_1 + \frac{d N_2}{dx} \varphi_2 \right ) dx = \int_0^l N_1 Q dx + q_0$

because $N_1(0)=1 \,$ and $N_1(l)=0 \,$

for i=2:

$\int_0^l \frac{d N_2}{dx} k \left ( \frac{d N_2}{dx} \varphi_1 + \frac{d N_2}{dx} \varphi_2 \right ) dx = \int_0^l N_1 Q dx - \overline q$

because $N_2(0)=0 \,$ and $N_2(l)=1 \,$

in matricial form:

$\begin{bmatrix} K_{11} & K_{12} \\ K_{21} & K_{22} \end{bmatrix} \begin{Bmatrix} \varphi_1 \\ \varphi_2 \end{Bmatrix} = \begin{Bmatrix} f_1 \\ f_2 \end{Bmatrix}$

with:

$K_{ij}=\int_0^{l} \frac{d N_i(x)}{dx} k \frac{d N_j(x)}{dx} dx$

$f_{1}=\int_0^{l} N_1(x) Q dx + q_0 \,$

$f_{2}=\int_0^{l} N_2(x) Q dx - \overline q$

To be more practical, this system of equations can be particularised for some specific polynomial shape functions:

$\begin{align} N_1^{(e)}(x) = \frac{x_2^{(e)} - x}{l^{(e)}} & \qquad \frac{d N_1^{(e)}(x)}{dx} = -\frac{1}{l^{(e)}} \\ N_2^{(e)}(x) = \frac{x - x_1^{(e)}}{l^{(e)}} & \qquad \frac{d N_2^{(e)}(x)}{dx} = \frac{1}{l^{(e)}} \end{align}$

Therefore:

$\begin{align} K_{11}^{(e)} &= K_{22}^{(e)} = - K_{12}^{(e)} = - K_{21}^{(e)} = \frac{k}{l^{(e)}} \\ f_1 &= \frac{Q l}{2} - q_0 \\ f_2 &= \frac{Q l}{2} + \overline q \end{align}$

$\frac{k}{l} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \begin{Bmatrix} \varphi_1 \\ \varphi_2 \end{Bmatrix} = \begin{Bmatrix} \frac{Q l}{2} - q_0 \\ \frac{Q l}{2} + \overline q \end{Bmatrix}$

$\begin{align} \varphi_1 & = \overline \varphi\\ \varphi_2 & = \frac{Q l^2}{2 k} - \frac{\overline q l}{k} + \overline \varphi \end{align}$

$\varphi_2 \,$ is the same value which it has been obtained with the analytical solution. Nevertheless, the global solution is clearly different:

$\hat \varphi (x) = \frac{l - x}{l} \varphi_1 + \frac{x}{l} \varphi_2 = \overline \varphi + \frac{1}{k} \left ( \frac{Q l}{2} - \overline q \right ) x \ne - \frac{Q}{2 k} x^2 + \frac{- \overline q + Q l}{k} x + \overline \varphi = \varphi(x)$

To obtain the reaction to the fixed $\overline \varphi \,$ value, the first equation of the matricial system of equations can be used:

$\frac{k}{l} (\varphi_1 - \varphi_2) = \frac{Q l}{2} + q_0 \qquad \Rightarrow \qquad \frac{k}{l} \left( \overline \varphi - \left( \frac{Q l^2}{2 k} - \frac{\overline q l}{k} + \overline \varphi \right) \right) = \frac{Q l}{2} + q_0$

$q_0 = \overline q - Q l \qquad \Rightarrow \qquad \underbrace{ q_0 + Q l }_{inflow} - \underbrace{ \overline q }_{outflow} = 0$

Resolution using two elements with two nodes

Two elements with two nodes each means the use of three global nodes:

$\hat \varphi (x) = N_1 (x) \varphi_1 + N_2 (x) \varphi_2 + N_3 (x) \varphi_3$

Locally, it is equivalent to:

$\left . \begin{align} N_1 &= N_1^{(1)} \\ N_1 &= 0 \end{align} \right \} \quad \begin{align} 0 \le & x \le l/2 \\ l/2 \le & x \le l \end{align}$

$\left . \begin{align} N_2 &= N_2^{(1)} \\ N_2 &= N_1^{(2)} \end{align} \right \} \quad \begin{align} 0 \le & x \le l/2 \\ l/2 \le & x \le l \end{align}$

$\left . \begin{align} N_3 &= 0 \\ N_3 &= N_2^{(2)} \end{align} \right \} \quad \begin{align} 0 \le & x \le l/2 \\ l/2 \le & x \le l \end{align}$

$\int_0^l \frac{d N_i}{dx} k \left ( \frac{d N_1}{dx} \varphi_1 + \frac{d N_2}{dx} \varphi_2 + \frac{d N_3}{dx} \varphi_3\right ) dx = \int_0^l N_i Q dx + \left[ N_i q \right ]_{x=0} - \left[ N_i \overline q \right ]_{x=l} \qquad i=1, 2, 3$

for i=1:

$\int_0^{l/2} \frac{d N_1^{(1)}}{dx} k \left ( \frac{d N_1^{(1)}}{dx} \varphi_1 + \frac{d N_2^{(1)}}{dx} \varphi_2 \right ) dx = \int_0^{l/2} N_1^{(1)} Q dx + \left[ N_1^{(1)} q \right ]_{x=0} - \left[ N_1^{(1)} \overline q \right ]_{x=l} = \underbrace{\int_0^{l/2} N_1^{(1)} Q dx }_{f_1^{(1)}} + q_0$

for i=2:

$\int_0^{l/2} \frac{d N_2^{(1)}}{dx} k \left ( \frac{d N_1^{(1)}}{dx} \varphi_1 + \frac{d N_2^{(1)}}{dx} \varphi_2 \right ) dx + \int_{l/2}^{l} \frac{d N_1^{(2)}}{dx} k \left ( \frac{d N_1^{(2)}}{dx} \varphi_2 + \frac{d N_2^{(2)}}{dx} \varphi_3 \right ) = \underbrace{\int_0^{l/2} N_2^{(1)} Q dx }_{f_2^{(1)}} + \underbrace{\int_{l/2}^{l} N_1^{(2)} Q dx }_{f_1^{(2)}}$

for i=3:

$\int_{l/2}^{l} \frac{d N_2^{(2)}}{dx} k \left ( \frac{d N_1^{(2)}}{dx} \varphi_2 + \frac{d N_2^{(2)}}{dx} \varphi_3 \right ) = \int_{l/2}^{l} N_2^{(2)} Q dx + \left[ N_2^{(2)} q \right ]_{x=0} - \left[ N_2^{(2)} \overline q \right ]_{x=l} = \underbrace{\int_{l/2}^{l} N_2^{(2)} Q dx }_{f_2^{(2)}} - \overline q$

in matricial form:

$\underbrace{ \begin{bmatrix} K_{11}^{(1)} & K_{12}^{(1)} & 0 \\ K_{21}^{(1)} & K_{22}^{(1)} + K_{11}^{(2)} & K_{12}^{(2)} \\ 0 & K_{21}^{(2)} & K_{22}^{(2)} \end{bmatrix} }_{K} \underbrace{ \begin{Bmatrix} \varphi_1 = \overline \varphi \\ \varphi_2 \\ \varphi_3 \end{Bmatrix} }_{a} = \underbrace{ \begin{Bmatrix} f_1^{(1)} + q_0 \\ f_2^{(1)} + f_1^{(2)} \\ f_2^{(2)} - \overline q \end{Bmatrix} }_{f}$

with:

$K_{ij}^{(e)}=\int_{l^{(e)}} \frac{d N_i^{(e)}(x)}{dx} k \frac{d N_j^{(e)}(x)}{dx} dx$

$f_{i}^{(e)}=\int_{l^{(e)}} N_i^{(e)}(x) Q dx \,$

In this case:

$\begin{align} K_{11}^{(e)} &= K_{22}^{(e)} = - K_{12}^{(e)} = - K_{21}^{(e)} = \frac{k}{l}^{(e)} \\ f_1^{(e)} &= \frac{Q l}{2} \\ f_2^{(e)} &= \frac{Q l}{2} \end{align}$

$\begin{bmatrix} \left( \frac{k}{l} \right ) ^{(1)} & -\left( \frac{k}{l} \right ) ^{(1)} & 0\\ -\left( \frac{k}{l} \right ) ^{(1)} & \left( \frac{k}{l} \right ) ^{(1)} + \left( \frac{k}{l} \right ) ^{(2)} & -\left( \frac{k}{l} \right ) ^{(2)} \\ 0 & - \left( \frac{k}{l} \right ) ^{(2)} & \left( \frac{k}{l} \right ) ^{(2)} \end{bmatrix} \begin{Bmatrix} \varphi_1 \\ \varphi_2 \\ \varphi_3 \end{Bmatrix} = \begin{Bmatrix} \frac{Q l^{(1)}}{2} + q_0 \\ \frac{Q l^{(1)}}{2} + \frac{Q l^{(2)}}{2} \\ \frac{Q l^{(2)}}{2} - \overline q \end{Bmatrix}$

If $l^{(e)}=l/2 \,$

$\frac{2 k}{l} \begin{bmatrix} 1 & -1 & 0\\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix} \begin{Bmatrix} \varphi_1 \\ \varphi_2 \\ \varphi_3 \end{Bmatrix} = \begin{Bmatrix} \frac{Q l}{4} + q_0 \\ \frac{Q l}{2} \\ \frac{Q l}{4} - \overline q \end{Bmatrix}$

$\begin{align} \varphi_1 & = \overline \varphi\\ \varphi_2 & = \frac{3 Q l^2}{8 k} - \frac{\overline q l}{2 k} + \overline \varphi \\ \varphi_3 & = \frac{Q l^2}{2 k} - \frac{\overline q l}{k} + \overline \varphi \end{align}$

and again $q_0 = \overline q - Q l \,$

Come back to the Resolution of the Poisson's equation using local Shape Functions

Resolution of the 1D Poisson's equation using local Shape Functions

Resolution using 1 element with two nodes

Resolution using two elements with two nodes

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