lagrange multiplier and contact elements

Hello,

I would like to know how contact elements are different from " general "ie non contact finite elements.

Also I would like to know how Lagrange multiplier method is used in contcat analysis.

My understanding of the Lagrange multiplier is like this: when we have to impose a constraint equation

[C][D]=[Q] as constraint between DOFs, we incude a term Y*{[[C][D]-[Q]} where Y is the Lagrange multiplier ,in the potential energy functional and minimize the functional  to get the algebraic equations.

My understadning is: during  contact analysis , we take an assumed state of contact and hence a constraint equation between the contacting and contacted nodes and the Lagrange method is imposed in the same way as above where this time [D] stands for the DOF which are "assumed" to contact.

Kindly advise if my understanding is correct.

Regards