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Can someone explain the strange physical behavior in contact problem using Lagrange method in modeling a simple 2D indentation?

I am modelling a very simple 2D contact problem between a rigid indenter and a deformable squared-shape specimen. I used a implicit function f to describe the rigid indenter. 
The contact condition is: Inside the contact zone of the deformable specimen, a node n is outside of rigid indenter for f(n)>0, and inside for f(n)<0. In case f(n)=0, the contacting node lies on the surface of the rigid body.
Then, I implemented the above contact condition for both Penalty and Lagrange method and make a comparison between two outputs: the deformed configuration of the Penalty method looks very reasonable in physical behavior but the Lagrange method seems to stick more strongly than enough to the indenter.

Does someone have idea or experience about this before? Could you please explain it to me? I attached the 2 deformed configuration of the 2 methods in order to depict my outputs. 

Comments

I think the penalty method works very good. Maybe you can check the force of Lagrange method, and compare the force and frictional condition of both methods. It seems that, the contact force is pulling the body to the edge! Errors could happen every where, it is not easy to find it directly, but just do it step by step.

Thank you very much for your reply. Actually, You are right, I fixed my bug by adding one more additional condition about contact pressure following the idea of Peter Wriggers in page 286 of Computational Contact Mechanics book. It is working fine now. I have already made a code2code comparison with ABAQUS. :-)

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