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?


1 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 $F$.

  • $\begingroup$ Dear Prof. Wolfgang Bangerth, @WolfgangBangerth You wrote a question for yourself and wrote an answer for it. I am trying to understand Lagrangian multipliers here. Your answer puts other readers under the presumption that this question has been answered by Mr. #1. So no one attempts it. Maybe you could have answered my question and then added your suggestion as extra information. Honestly, you are not being helpful at the moment. Please don't take any question as dumb. Maybe they have a reason. Be open-minded. Don't answer questions if you think they are too simple for you. My kind request. $\endgroup$ Commented Jun 24, 2021 at 17:06
  • $\begingroup$ Dear Prof @WolfgangBangerth, You did not even answer your own question fully. You say "It requires a bit of work". "You just need to differentiate various occurrences". So the next time, you'll write "Yes. I know the answer to this. It's easy"? Everyone knows that it involves differentiation. Why do you feel so compelled to write something when you are really not interested in helping someone? I have a feeling that you are just trying to show that you are smart. Please refrain from writing incomplete answers. As I said it's because I don't think people will write after you write. $\endgroup$ Commented Jun 24, 2021 at 17:34
  • $\begingroup$ @BruceLeeJunFan First of all, I am sure that Prof. Bangerth has better things to do than "showing that he is smart". Second, he answered to your question: it concerned the nonlinear term, so taking into account the Lagrange multipliers is indeed unnecessary in this case. "Everyone knows that it involves differentiation.": then you should precisely ask what you want to know. "I don't think people will write after you write.": then you do not know how the StackExchange sites work. If someone finds the question interesting and has something to add, they will write. $\endgroup$ Commented Jun 25, 2021 at 12:09
  • $\begingroup$ @ZoltánCsáti I myself said that the Prof has better things to do. Also, I am not interested in entertaining dialogue with you if you are only here to please Prof Bangerth for a fellowship. However, differences aside, I'd certainly appreciate an elaborate answer with all due respect. I respect sincere efforts even if they don't help. Expecting your kind cooperation in this regard. $\endgroup$ Commented Jun 25, 2021 at 13:37
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    $\begingroup$ @BruceLeeJunFan I do have better things to do. Since you think that my answer -- written in earnest belief that that is what you were asking -- does not satisfy you, I offer you to simply ignore it. $\endgroup$ Commented Jun 26, 2021 at 4:37

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