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

Best,

J.

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.