The method for solving nonlinear eqation systems implemented in most FEA packages is the Newton Method or Newton-Cotes Method. It converges well in most cases, but sometimes (if the initial guess for the solution is “far” from the real solution, or if the tangent matrix is not updated) the convergence may not be reached and the solver diverges.
In those cases it might be useful to use the line search method to stabilize the convergence and sometimes also reduce the number of iterations required to reach convergence.
If we denote with \(\delta u_n^i\) the displacement update within a given Newton increment \(n\) and iteration \(i\), the line search will try to minimize the residual with a variable pitch of advance \(\rho \in [0,1]\) in the current search direction \(\delta u_n^i\) and update the displacements according to:
If the line search is activated the user can select between two different methods:
- Secant method:
The secant method is a well known and simple one-dimensional search algorithm.
- Mixed method:
The mixed method is a variant of the secant method with variable bounds for \(\rho\)
For every method there are two parameters to be specified by the user:
This parameter specifies the threshold of the residual below the convergence of the line search is assumed. A relatively large residual of 0.1 should be a good choice for the convergence threshold, since we only want to improve the Newton solution.
- Maximum Iterations:
This is the maximum number of iterations for the line search to reach convergence. Experience shows that a low number of iterations (2-3) is enough for the secant method to reach convergence, whereas the mixed method might need up to 10 iterations.
It is not recommended to use line search in simulations where physical contact is active.