Assembling mass matrix for transient heat conduction pde

[nu_hash]

I have derived the weak form of the transient heat conduction equation (for FEM) and I am having trouble trying to assemble the mass matrix

This is the PDE:
$\frac{\partial U}{\partial t} = \alpha \nabla^2U$

This is the equation for the mass matrix for an element:
$M^e = \int \Psi \Psi^T dx$
where psi is a matrix containing the shape functions of the element.

I am quite new to FEM so I am not sure how the mass matrix is supposed to be assembled, I understand that I have to use the gauss quadrature to complete the integral but I just can't figure how the matrix containing the shape functions is assembled (i.e. I don't know what numbers go where)

Any help would be greatly appreciated.

 

[AlephZero]

$\mathbf\Psi$ is a vector $\begin{bmatrix}\psi_1(x) \\ \cdots \\ \psi_n(x) \end{bmatrix}$ where the $\psi_i(x)$ are the shape functions for the $n$ nodes (grid points) in the element.

So $\mathbf \Psi\mathbf\Psi^T$ is an $n \times n$ matrix, and you integrate the individual terms $M_{ij}^e = \int \psi_i(x)\psi_j(x)\,dx$.

To assemble the element matrices into the global matrix, you label the rows and columns of the global matrix using the node (grid) numbers, and then add each element matrix into the the correct node number positions.

 

[nu_hash]

Thanks that seems to make more sense now.