Hi I am trying to implement a simple plasticity based finite element code. I am not clear how to set up displacement control applied through Lagrange multipliers. In case of a linear problem, I did the following to apply displcament control and it worked. Lets say the energy balance equation in the absence of body force is $$ \frac{1}{2} \bf K u^2=f_{ext}u $$ Now I need to apply boundary conditions at some dof , $u_b$. The contraint equations can be written as $$ Au_b=u_{app}, $$ where, $u_b$ are the constrained/controlled DOF. $u_{app}$ are the constrain values. $\lambda$ - pseudo load term. The combined energy functional is $$ \frac{1}{2} \bf K u^2 + \lambda(Au_b-u_{app)}=f_{ext}u $$ If I differentiate the above energy term w.r.t $u$ and $\lambda$, I get the following force balance set as below $$ \bf K \bf u + \lambda A= \bf f_{ext} $$
$$ \begin{bmatrix} \bf K & \bf A\\ \bf A^T & \bf 0 \end{bmatrix} \begin{Bmatrix} \bf u\\ \lambda \end{Bmatrix}= \begin{Bmatrix} \bf f_{ext}\\ u_{app} \end{Bmatrix} $$ Incase of non-linear problems, the $\bf K$ is $\bf K(u)$ and is non-linear. So the energy balance equation is $$ \frac{1}{2} \bf K(\bf u) u^2 + \lambda (A \bf u_b-\bf u_{app})=\bf f_{ext}\bf u $$ If I differentiate the above energy term w.r.t $u$, I get the force balance as below $$ \frac{1}{2} \frac{\partial{\bf K(\bf u)}}{\partial{\bf u}} \bf u^2 + \bf K \bf u + \lambda \bf A= \bf f_{ext} $$ It feels wrong. How do I evaluate this? How are the matrix form of equations modified in the non-linear case?