Dear Samson

the FEM code can be written by MATLAB in the use of programming. for programming, the flowing subroutine must be considered:

1- main program: in this page, all of the subroutine must be called.

2- write data: in the use of this subroutine, a input file can be prepared.

3- input data: the input file readied using this sub-program.

4- form the stiffness matrix

5- assemble the local stiffness matrix in the global stiffness matrix

6- form the force vector.

7- Applying the boundary conditions in the global stiffness in order to solve the equation.

8- Solve the equation.

9- Post processing for the program...

best regards

Hope  this post would be helpful: https://femexpert.wordpress.com/2015/03/15/2d-fem-matlab-contact/

Your weak form does not have the interface term for contact. The weak form should include impenetrability constraint in the form of penalty or Lagrange multiplier.

Yes, the summation implies assembly. For one pair of a slave node and a master element, the coupled DOFs can be written as {x_s, x_m1, x_m2, ...}. Therefore, during assembly, the corresponding force terms are added to both the slave node and the nodes in the master element.