Implicit dynamic solver using non-linear Newmark's method with example file

function Result=Newmark_Nonlinear(Elements,Material,Support,Free,M,C,f,fs,delta)

Input
Elements: a structure containing Elements{i}.DOFs and Elements{i}.Material
where Elements{i}.DOFs=[j k] means element i connect DOF j with k
and Elements{i}.Material=m assign material m to element i

Material: a structure containing material properties for bilinear springs
where Material{m}.k1 is Spring stiffness
Material{m}.x1 is Spring deformation beyond which the stiffness decreases
Material{m}.k2 is Reduced stiffness

Support: a vector of support (Fixed) DOFs of size (nSupport,1)
Free: a vector of free DOFs of size (nFree,1)
M:mass matrix (nFree*nFree)
C:damping matrix (nFree*nFree)
f:external force matrix(nFree,N)
fs: sampling frequency
delta: convergance criterion for residual force
where N is the length of data points of dynamic force

Output:
Result: is a structure consist of
Result.Displacement: Displacement (nFree*N)
Result.Velocity: Velocity (nFree*N)
Result.Acceleration: Acceleration (nFree*N)

Note: Elements are assumed to be springs connecting nodes with bi-linear stiffness (No hysteresis).

References
Chopra, Anil K. "Dynamics of Structures. Theory and Applications to." Earthquake Engineering (2017).