# FEM, Direct Stiffness Method with a nonlinear displacement constraint in one node

i have a question about a FE problem im working on. I made a finite element model of an linear elastic block of material (double striped block) attached with a rigid connection to the environment (colored block). A force is applied in the bottom right node.

The nodal displacements are calculated according to the Direct Stiffness Method. The system of equations is simply KU = F or U = CF with C=K^-1.

The displacement of the top right node needs to be constrained to a specified trajectory. By adding lagrange multipliers to the global stiffness matrix i have succesfully constrained the possible displacements of the top right node to a line. To be clear the force is the input, the displacement is the output.

The new system of equations is:

I want to extend the problem and therefore it is needed for me to specify a nonlinear trajectory. As far as i know the direct stiffness method with lagrange multipliers does not work in this case for it only allows to add constraints of the form y = a*x+b.

What is the best method to add a nonlinear displacement constraint to this system of linear equations?

I have been searching and reading papers in the subject but so far i have not found a sound answer.

I hope you guys can help me and i am lloking forward to your answer,

Best,

J.

## 1 Answer

The key point to recognize is that the Lagrange multiplier method is applicable to both nonlinear and linear constraints.

Assuming you can write your nonlinear constraint equation as $$g(U_s) = 0$$ you have a set of N+1 nonlinear equations to solve as follows:

$$K_{ss}U_s + \lambda\frac{\partial g}{\partial U_s} = F_s$$ $$g(U_s) = 0$$ where N is the number of finite element equations.

These can be solved for $U_s$ and $\lambda$ using the Newton-Raphson method. If you are not already familiar with using Newton-Raphson for solving a system of nonlinear equations, you can find many references on the web.

Introductory FEM courses almost always emphasize linear formulations and so present the Lagrange multiplier technique as you show above. In intermediate rigid body dynamics courses, in either physics or engineering departments, the equations of motion and constraints are almost always nonlinear, so the Lagrange multiplier method is presented from that point of view. So, if you want to learn more, I suggest taking a look at a dynamics text used in such a course.

• Would you by any chance have a link to such a intermediate course? PS. Thanks for your reply!! Commented May 25, 2016 at 15:18
• The Wikipedia article on this topic is pretty good: en.wikipedia.org/wiki/…. This text has been used for engineering courses for many years and is quite readable: amazon.com/Principles-Dynamics-2nd-Donald-Greenwood/dp/… Commented May 25, 2016 at 16:12