Your Task

Write the function assembleKff$()$ that takes five input arguments n$\_$dof, n$\_$elem, elem$\_$node, elem$\_$stiff$\_$val, eq$\_$num and returns one output k$\_$ff:

Input arguments:

n$\_$dof: an integer representing the number of nodes with free displacement $($not prescribed$).$ This is the largest positive number in eq$\_$num.

n$\_$elem: an integer representing the number of spring elements.

elem$\_$node: a $2-$D NumPy array of size $(\_2)($n$\_$elem$2)$ containing the left$/$right node numbers for all string elements.

elem$\_$stiff$\_$val: a $1-$D NumPy array of length $\_$n$\_$elem containing the stiffness values for all string elements, matching the order of elem$\_$node.

eq$\_$num: a $1-$D NumPy array of length $\_$n$\_$node that determines how you should reorder your system of equations.

Output:

k$\_$ff: a $2-$D NumPy array that represents the global stiffness matrix corresponding to the free nodes.

Algorithm to create the array k$\_$ff:

Initialize the k$\_$ff array with zeros. Make sure to use the correct type. What should the size of k$\_$ff be$?$

Loop over all spring elements in elem$\_$node. For each element:

Create the element stiffness matrix $$kell$.$Identify the four node pairs for the element.Check if both nodes in a pair are free nodes. Only if both nodes in a pair are free can that pair contribute to the global matrix k$\_$ff$.$add the contributions of $$kell into the global matrix k$\_$ff$,$ following the procedure described above.