Specific Examples of the Resolution of the Poisson's equation with the WRM using global Shape Functions

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The matrix form of the residual formulation can be written as: \mathbf{K} \cdot \mathbf{a} = \mathbf{f} with:


\begin{cases} K_{ij} & = \int_{\Omega} W_i(x) \left[ \frac{d}{dx} k \frac{d}{dx} \left( N_j (x) \right) \right] d\Omega \\ f_i & = \frac{}{} \int_{\Omega} W_i(x) Q(x) d\Omega \end{cases}


For simplicity reasons, we select as basis functions the Fourier Series (N_i(x)=\sin \frac {\pi x i}{l} \,), that automatically accomplish the boundary conditions (N_i(0)=0 \, and N_i(x_L)=0 \,). If the material is constant, we can write:


\frac{d}{dx} k \frac{d}{dx} \left ( \sin \frac {\pi x i}{l} \right) = \left ( - k \left (\frac{\pi i}{l} \right )^2 \sin \frac {\pi x i}{l} \right) = - k \left (\frac{\pi i}{l} \right )^2 \sin \frac {\pi x i}{l}


\begin{cases} K_{ij} & = - \int_{\Omega} W_i(x) k \left (\frac{\pi j}{l} \right )^2 \sin \frac {\pi x j}{l} d\Omega \\ f_i & = - \frac{}{} \int_{\Omega} W_i(x) Q(x) d\Omega \end{cases} \Rightarrow \begin{cases} K_{ij} & = \left (\frac{\pi j}{l} \right )^2 \int_{\Omega} W_i(x) k \sin \frac {\pi x j}{l} d\Omega \\ f_i & = \frac{}{} \int_{\Omega} W_i(x) Q(x) d\Omega \end{cases}


sources medium boundary

conditions

shape functions
constant source
homogeneous
Dirichlet
Fourier Series
Q(x) = \begin{cases} Q_0 & x_0 \leqslant x \leqslant x_f \\ 0 & x_f \leqslant x \leqslant x_L \end{cases}
k(x) = cte \frac{ }{ }
\varphi (x_0)= \varphi_0
\varphi (x_L)= \varphi_L
N_i(x)=\sin \frac {\pi x i}{l} \,


By using different weighting functions, a new range of methods can be classified:


Contents

Collocation Method     (\frac{}{} W_i(x)=\delta(x - x_i), \frac{}{} i=1, 2,... , n)

General expression k and Q constant, using Fourier series
\begin{cases} K_{ij} & = \frac{d}{dx} k \frac{d}{dx} N_j (x) \bigg|_{x=x_i} \\ f_i & = Q(x_i) \, \end{cases}
\begin{cases} K_{ij} & = k \left (\frac{\pi j}{l} \right )^2 \sin \frac {\pi x_i j}{l} \\ f_i & = Q(x_i) \, \end{cases}


Number of points = 1
\begin{cases} x_1 & = \frac{l}{2} \\ Q(x_1) & = \frac{Q}{2} \\ K_{11} & = k \left (\frac{\pi}{l} \right )^2 \sin \frac {\pi x_1}{l} \\ f_1 & = Q(x_1) \, \end{cases}


Number of points = 2
\begin{cases} x_1 & = \frac{l}{4} \\ x_2 & = \frac{3 l}{4} \\ Q(x_1) & = Q \\ Q(x_2) & = 0 \\ K_{ij} & = k \left (\frac{\pi j}{l} \right )^2 \sin \frac {\pi x_i j}{l} \\ f_i & = Q(x_i) \, \end{cases}


Number of points = n
\begin{cases} x_i & = \frac{i l}{n+1} \\ Q(x_i) & = \begin{cases} Q & if ~ x < \frac{l}{2} \\ 0 & if ~ x > \frac{l}{2} \end{cases} \\ K_{ij} & = k \left (\frac{\pi j}{l} \right )^2 \sin \frac {\pi x_i j}{l} \\ f_i & = Q(x_i) \, \end{cases}


CollocationWRM.jpg
To modify the parameters, edit this Matlab code

Subdomain Method     (W_i(x)=1 \quad \forall x \in \Omega_i \quad and \quad W_i(x)=0 \quad \forall x \notin \Omega_i )

General expression k and Q constant, using Fourier series
\begin{cases} K_{ij} & = \int_{\Omega_i} \left[ \frac{d}{dx} k \frac{d}{dx} \left( N_j (x) \right) \right] d\Omega \\ f_i & = \frac{}{} \int_{\Omega_i} Q(x) d\Omega \end{cases}
\begin{cases} K_{ij} & = k \left (\frac{\pi j}{l} \right )^2 \int_{\Omega_i} \sin \frac {\pi x j}{l} d\Omega = - k \left (\frac{\pi j}{l} \right ) \cos \frac {\pi x j}{l} \Big|_{\Omega_i} \\ f_i & = Q(x) \Big|_{\Omega_i} \int_{\Omega_i} d\Omega \, \end{cases}



Number of points = 1
\begin{cases} x_1 & = \frac{l}{2} \\ Q(x_1) & = \frac{Q}{2} \\ K_{11} & = 2 k \left (\frac{\pi}{l} \right ) \\ f_1 & = Q(x_1) \, \end{cases}


Number of points = 2
\begin{cases} x_1 & = x_0 \\ x_2 & = \frac{l}{2} \\ x_3 & = x_L \\ Q(x_1) & = Q \\ Q(x_2) & = 0 \\ K_{ij} & = - k \left (\frac{\pi j}{l} \right ) \left ( \cos \frac {\pi x_{i+1} j}{l} - \cos \frac {\pi x_{i} j}{l}\right )\\ f_i & = Q(x_i) (x_{i+1} - x_i)\, \end{cases}


Number of points = n
\begin{cases} x_1 & = x_0 \\ x_{i+1} & = \frac{i l}{n} \\ Q(x_i) & = \begin{cases} Q & if ~ x < \frac{l}{2} \\ 0 & if ~ x > \frac{l}{2} \end{cases} \\ K_{ij} & = - k \left (\frac{\pi j}{l} \right ) \left ( \cos \frac {\pi x_{i+1} j}{l} - \cos \frac {\pi x_{i} j}{l}\right )\\ f_i & = Q(x_i) (x_{i+1} - x_i) \, \end{cases}


SubdomainWRM.jpg
To modify the parameters, edit this Matlab code

Galerkin Method     (\frac{}{} W_i(x) \equiv N_i(x))

General expression k and Q constant, using Fourier series
\begin{cases} K_{ij} & = \int_{\Omega} N_i(x) \left[ \frac{d}{dx} k \frac{d}{dx} \left( N_j (x) \right) \right] d\Omega \\ f_i & = \frac{}{} \int_{\Omega} N_i(x) Q(x) d\Omega \end{cases}
\begin{cases} K_{ij} & = \int_{\Omega} \sin \frac {\pi x i}{l} k \left (\frac{\pi j}{l} \right )^2 \sin \frac {\pi x j}{l} d\Omega \\ & = k \left (\frac{\pi j}{l} \right )^2 \int_{\Omega} \sin \frac {\pi x i}{l} \sin \frac {\pi x j}{l} d\Omega \\ f_i & = \frac{}{} \int_{\Omega} \sin \frac {\pi x i}{l} Q(x) d\Omega \end{cases}




Number of points = 1
\begin{cases} x_1 & = \frac{l}{2} \\ Q(x_1) & = Q \, \\ K_{11} & = k \left ( \frac{\pi^2}{2 l} \right) \\ f_1 & = \left ( \frac{Q(x_1) l}{\pi} \right)\, \end{cases}


Number of points = 2
\begin{cases} x_1 & = x_0 \, \\ x_2 & = \frac{l}{2} \\ x_3 & = x_L \, \\ Q(x_1) & = Q \\ Q(x_2) & = 0 \\ K_{ij} & = \begin{cases} k \left (\frac{\pi j}{l} \right )^2 \frac {l}{2} & if ~ i = j \\ k \left (\frac{\pi j}{l} \right )^2 \frac {l}{2 \pi} \left( \frac {sin(\pi (i-j))}{i-j} - \frac {sin(\pi (i+j))}{i+j} \right) & if ~ i <> j \end{cases} \\ f_i & = Q l \frac{1 - cos(\frac{i \pi}{2})}{i \pi} \, \end{cases}


Number of points = n
\begin{cases} x_1 & = x_0 \, \\ x_{i+1} & = \frac{i l}{n} \\ Q(x_i) & = \begin{cases} Q & if x < \frac{l}{2} \\ 0 & if x > \frac{l}{2} \end{cases} \\ K_{ij} & = \begin{cases} k \left (\frac{\pi j}{l} \right )^2 \frac {l}{2} & if ~ i = j \\ k \left (\frac{\pi j}{l} \right )^2 \frac {l}{2 \pi} \left( \frac {sin(\pi (i-j))}{i-j} - \frac {sin(\pi (i+j))}{i+j} \right) & if ~ i <> j \end{cases} \\ f_i & = Q l \frac{1 - cos(\frac{i \pi}{2})}{i \pi} \, \end{cases}


GalerkinWRM.jpg
To modify the parameters, edit this Matlab code

Least Squares Method     (W_i(x)=A(\varphi) \quad and \quad \overline{W_i}(x)=B(\varphi))

General expression k and Q constant, using Fourier series
\begin{cases} K_{ij} & = \int_{\Omega} 2 \left[ \frac{d^2 N_i(x)}{dx^2} \right] \left[ \frac{d}{dx} k \frac{d}{dx} \left( N_j (x) \right) \right] d\Omega \\ f_i & = \frac{}{} \int_{\Omega} 2 \left[ \frac{d^2 N_i(x)}{dx^2} \right] Q(x) d\Omega \end{cases}
\begin{cases} K_{ij} & = - \int_{\Omega} 2 \left( \frac{\pi i}{l} \right)^2 N_i(x) \left[ \frac{d}{dx} k \frac{d}{dx} \left( N_j (x) \right) \right] d\Omega \\ & = - 2 \left( \frac{\pi i}{l} \right)^2 \int_{\Omega} N_i(x) \left[ \frac{d}{dx} k \frac{d}{dx} \left( N_j (x) \right) \right] d\Omega \\ f_i & = - \int_{\Omega} 2 \left( \frac{\pi i}{l} \right)^2 \frac{}{} N_i(x) Q(x) d\Omega \\ & = - 2 \left( \frac{\pi i}{l} \right)^2 \frac{}{} \int_{\Omega} N_i(x) Q(x) d\Omega \end{cases}


Note that, in this case and without generality, due to the selected basis functions, the obtained solution is the same that the obtained one using the Galerkin method scaled with a factor 2 \left( \frac{\pi i}{l} \right)^2.



LeastSquaresWRM.jpg
To modify the parameters, edit this Matlab code




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