Solver

Creates an instance of a Solver

System class methods

method	input	returns	description
__init__(…)			constructor.
addNode(newNode)	Node(…) object		add one Node() object to your list of elements (the model)
addElement(newElem)	Element(…) object		add one Element() object to your list of elements (the model)
solve()			assemble \([K_t]\) and \(\{P\}\), solve for \(\{u\} = [K_t]^{-1}\{P\}\), loop through nodes and update nodal displacement, compute unbalanced force \(\{R\} = \{P\} - \{F\}\)
plot(factor=1.0)			collect node info and send it to the plotter. Request the plot.
report()			print a summary report: list of nodal position, load, displacement, unbalanced force.

System class variables

name	type	description
nodes	List of Node() objects	holds all the nodes in the model
elements	List of Element() objects	holds all the elements in the model
disp	np.array([…])	system sized displacement vector
loads	np.array([…])	system sized load vector

Equations

1. each element has node0 = elem.nodes[0] and node1 = elem.nodes[1]

2. node indices i = node0.index and j = node1.index

3. a local d.o.f. \(u\to k=0\) or \(v\to k=1\) of node \(i\) belongs at global index \(K = 4*i + k\)

4. a local d.o.f. \(u\to m=0\) or \(v\to m=1\) of node \(j\) belongs at global index \(M = 4*j + m\)

Assembly:

• \({\bf F}\): element force from elem.getForce()

• \({\bf K_t}\): element stiffness from elem.getStiffness()

• \({\bf R}_{sys}\): system force

• \({\bf K_t}_{sys}\): system stiffness

5. Loop over nodes: \({\bf R}_{sys}[K] = nodes[i].getLoad()[k]\) (this should return 0 if no load at this node and d.o.f.)

6. Loop over elements: \({\bf R}_{sys}[K] = {\bf R}_{sys}[K] - {\bf F}[i][k]\)

7. Loop over elements: \({\bf K_t}_{sys}[K,M] = {\bf K_t}_{sys}[K,M] + {\bf K_t}[i,j][k,m]\)

8. Loop over nodes: if a d.o.f. at K is fixed, set \({\bf R}_{sys}[K] = 0\) and \({\bf K_t}_{sys}[K,K] = 1.0e20\).

9. Solve system of equations: \({\bf U} = {\bf K_t}_{sys}^{-1}\,{\bf R}_{sys}\)

10. Loop over nodes: \(nodes[i].setDisp(u,v)\) using \(u = {\bf U}[2*i]\) and \(v = {\bf U}[2*i+1]\)

11. Recompute: \({\bf R}_{sys}\) as in steps 5 and 6 (do not repeat steps 7-11). If everything was done correctly, fixed d.o.f.s will contain the support reactions and free d.o.f.s should hold numeric zeros.