Applying displacement control loading using lagrange multipliers in the material non-linear finite element method

Question

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?

Answer 1

The addition of the Lagrange multiplier term is an unnecessary complication in this case, so let's just consider the simpler root finding problem $$ F(u) = \frac12 u^TK(u)u - fu = 0. $$ Your question is how to solve this using, for example, a Newton method. The answer is that you need to solve a sequence of problems of the form $$ J(u_k) \; \delta u_k = -F(u_k) $$ where the Jacobian matrix $J$ is defined as $$ J(u)_{ij} = \frac{\partial F(u)_i}{\partial u_j}. $$

It requires a bit of work to sort out how $J_{ij}$ looks like, but it's not conceptually difficult. You just need to differentiate the various occurrences of the terms in $F$ with regard to $u_j$ and keep track of the summations that appear in $F$.