In this lecture, we will cover the following:

• How to assemble the primary structure stiffness matrix, element by element,
• How to reduce the primary stiffness matrix down to the structure stiffness matrix by removing rows and columns associated with restrained degrees of freedom.

This lecture is really the heart of our finite element implementation. We start by importing everything from our utils file so that we have access to the stiffness matrix calculation function we built and saved at the end of the last section.

We then loop through every element and, for each one, extract its transformation matrix and its local $x$ and $y$ coordinates. We pass these into our stiffness matrix calculation function which returns the $20 \times 20$ local element stiffness matrix. We then transform this into a $24 \times 24$ global stiffness matrix using our previously calculated element transformation matrix.

To work out where in the primary stiffness matrix this contribution slots in for the current element, we calculate the relevant degrees of freedom for each node that define the element. With a little help from Numpy, we turn the element degree of freedom list into an index array that lets us directly add the element's contribution into the correct rows and columns of the primary stiffness matrix in a single line. Once we have looped through every element, we delete the rows and columns corresponding to restrained degrees of freedom to produce the reduced structure stiffness matrix, which is the version we will ultimately invert.

Next up

In the next lecture, we will build the global force vector for our structure using the self-weight of the shell.