I'm using the finite element method to obtain the time response of a structure harmonically excited. I'm using a linear displacement function to obtain the stiffness matrix and the consistent mass matrix, i.e., I have this set of equations:
$$[M]\{\ddot x\} + [K]\{x\} = \{F(t)\}$$
What is the best way to apply the boundary conditions on the $[M]$ and $[K]$ matrices?
Typically, I have found two ways:
- Remove the rows/columns whose degree of freedoms are constrained
- Modify the matrices setting to zero the rows and columns and placing ones on the diagonal to maintain non-singularity.
I prefer the second way, since we can adjust it to account for non zero displacement constrains.
The problem here is that I'm using the Newmark method to integrate the system and the biggest difficult is that we are dealing with geometrical nonlinearities, which means $[K]$ is the tangent stiffness matrix. Since this is a nonlinear problem, we modify the Newmark method to account for the nonlinearities and use the Newton-Raphson method to find the equilibrium position at time $t(i+1)$ when marching forward in time. Usually, the Newton-Raphson method is fast enough to find the equilibrium position. However, if the initial estimate is not good enough, it will not converge.
The biggest problem here is that, when modifying the matrices, an error residue on any degree of freedom found by the Newton-Raphson is propagated and the final answer diverges. Any ideas on how to go around this problem?
I'm trying to keep it simple, without too much details on the Newmark method or the FEM method. If necessary, however, we can discuss it.