# FEM: What's D in the equation for half band width

In the equation for half band width given below,

B = (D + 1)f

my textbook states D to be the maximum difference between node numbers in an element and all the elements are considered for its determination.

I don't understand what the above statement means, can someone please explain what D means and how to find its value.

• Can you add the reference of your textbook? Jun 13, 2018 at 15:00

This would only apply for a simple FE program where the nodes in the model are always labelled from 1 to $n$, every node has the same number of degrees of freedom $f$, and the rows and columns of the global matrix are also ordered in groups of $f$ from 1 to $n$.
First, think about the case where there is only one degree of freedom at each node, or $f = 1$. If the smallest and largest notes in an element are $p$ and $q$, when you assemble element matrix into the global matrix all the terms fit within the square submatrix between rows and columns $p$ and $q$, so the maximum distance of the terms from the diagonal of the global matrix (i.e. the bandwidth) is $q-p+1$ or $D+1$ in your notation.
The bandwidth of the global matrix is therefore the maximum value of $D+1$ for all the elements in the model.
If there are $f$ degrees of freedom at each node, the system matrix is $f$ times bigger, and the bandwidth is also multiplied by $f$ to give your formula.
Note 1, the value of $D$ in each element depends how you number the nodes in the model, and reducing the bandwidth of the global matrix will reduce the computer time required to solve the problem.