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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