Contact problem, which exists widely in mechanical engineering, civil engineering, manufacturing engineering, etc., is an extremely complicated nonlinear problem. It is usually solved by the finite element method. Unlike with the traditional finite element method, it is necessary to set up contact elements for the contact analysis. In the different types of contact elements, the Goodman joint elements, which cover the surface of contacted bodies with zero thickness, are widely used. However, there are some debates on the characteristics of the attached elements of the Goodman joint elements. For that this paper studies the type, matching, and meshing of the attached elements. The results from this paper would be helpful for the finite element contact analysis.