Let's assume $F$ is an external load for a nonlinear static finite element problem. Normally, the problem will not converge if you apply $F$ fully. Instead, we multiply the load $F$ with a scaling factor $\lambda$, solve the problem with a small $\lambda$ and gradually increase it to $1$.
For example, assume we have a converged solution $u_n$ at step $n$ (with the scaling factor $\lambda_n$), we then solve the $n+1$ step by Newton-Raphson: $$\lambda_{n+1}=\lambda_{n}+\Delta\lambda$$ $$K(u_n)\Delta u = \lambda_{n+1}F$$ $$u_{n+1}=\Delta u+u_{n}$$
Currently, the step size $\Delta \lambda$ is a constant in my FEM implementation and I feel it is inefficient. Is there any algorithm that can determine the step size $\Delta \lambda$ automatically?
