# 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.? – nicoguaro Aug 19 '19 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. – Gustavo Costa Aug 19 '19 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. – nicoguaro Aug 19 '19 at 20:50

## 1 Answer

I agree that Crisfield's book is not a clear place to start for this topic. Have you read the original paper by Riks, "An incremental approach to the solution of snapping and buckling problems", IJSS 1979? It is quite clear in my opinion. If you don't have access, I can send it to you. I believe the modification by Crisfield is widely used, and his paper "A fast incremental/iterative solution procedure that handles 'snap-through'", Computers & Structures 1981 is also clearly written. If you have any specific questions about implementation I can try to help.

• OskarM, could you provide a step-by-step procedure? I was trying to solve a single equation as follows: [K(u)].{u} = {F} with the Stiffness Matrix as [K(u)] = u+1; Displacement vector {u} = u and Force vector {F} = 2. I guess by solving this very simple equation then the procedure for a system of nonlinear equations will be straightforward. – Gustavo Costa Aug 20 '19 at 3:14
• I'm not sure I understand your notation. I don't know what you mean to indicate by the square brackets in [K(u)], nor by the curly braces in {F}. I also don't know what you mean by [K(u)] = u+1, {u}=u, or {F} = 2. Can you try to explain a bit more what you mean? I would be happy to summarize Riks' method, as developed in his original paper, as well as the modification by Crisfield, but it will take more space than one comment. – OskarM Aug 20 '19 at 3:26
• [ ] is a matrix and { } a vector. Since this case is a single equation, [K(u)] is a 1x1 matrix; {u} and {F} are 1x1 vectors. K(u) is just to say that the Stiffness Matrix is a function of the unknowns "u", which means this is a nonlinear problem. Thus, I'm trying to solve u²+u = 2. – Gustavo Costa Aug 20 '19 at 3:32
• 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. – OskarM Aug 20 '19 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. – OskarM Aug 20 '19 at 3:54