# Reference request: Riks method (Nonlinear FEM)

I'm struggling to find a good detailed reference explaining the Arc-length method or, more generally, Riks method and its derivations. I looked for the classical books in nonlinear mechanics (the ones by Crisfield, De Borst, Bathe, etc) but they're confusing for a beginner in nonlinear analysis. Thus, I'd like to request for references regarding the Arc-length method for systems of nonlinear equations. I'm not new to FEM and I'm aware of Newton-Raphson method but the problem I'm dealing with presents a snap-through behavior due to softening of the material.

• Have you checked section 6.5.3 of Belytschko, Ted, et al. Nonlinear finite elements for continua and structures. John wiley & sons, 2013.? Commented Aug 19, 2019 at 18:27
• @nicoguaro yes and it still seems a bit tough to me regarding implementation. I was looking for a reference showing how to solve it at least for a single equation, something like a step-by-step procedure.
– user30551
Commented Aug 19, 2019 at 20:30
• Well, I have seen material like that for linear analysis but never for the nonlinear counterpart. But, I hope I'm wrong, and there is something like that. Commented Aug 19, 2019 at 20:50

• OK, I think I see what you mean - you want to apply Riks' method to a problem with a single degree of freedom for the displacement variable. You have two variables - the loading, and the scalar displacement. I'm still not sure about your terminology. The general problem is usually described as a nonlinear system, $F(u)=0$. In elasticity, $F$ is a force, which can be decomposed into applied loads and internal stresses. Then the problem is linearized to get $K(u) du + F(u) = 0$, where $K$ is the stiffness matrix/Jacobian of F, and $du$ is the incremental displacement, which is solved for. Commented Aug 20, 2019 at 3:44
• Anyway, I'm working on an edit to include a step-by-step procedure but I probably won't get to it tonight. The basic idea of Riks' method is to add a "control" equation to govern the loading parameter. For spherical arc-length control, you would have the equation $d\lambda^2+|du|^2=R^2$, where $d\lambda$ is the load increment and $R$ is the step size, which is typically tuned to get efficient convergence of Newton's method. There is no equation that looks like $u^2+u=2$, unless I'm missing something specific to your question. Commented Aug 20, 2019 at 3:54