Investigation of Mechanical Contact Problems with High-order Finite Element Methods

The current study deals with the emerging field of mechanical contact problems. The behavior of two dimensional uni- and bilateral, frictionless and frictional contact problems is studied by using the h-, p-, hp-, and the rp-version of the Finite Element Method. The contact constraints are introduced by the penalty method. The main focus of the work is drawn to the level of accuracy which is to be derived with the different versions of the FEM. The accuracy is measured for the unilateral Hertzian contact problem in terms of the contact stress distribution in the contact interface and its neighborhood. Additionally, the convergence of the error in strain energy norm is studied as well. Both studies prove the capability of the p-version of the Finite Element Method and its adaptive extensions. A high polynomial degree in combination with an exact representation of the curved geometry leads together with the rp-method to a very high accuracy in the contact stresses. This is achieved by relocation of a Finite Element node to the end of the contact interface where the boundary condition changes from contact to no-contact, avoiding oscillations in the high-order polynomial shape functions at this point. The convergence rate of the strain energy reveals that a singularity exists in the solution domain. This singularity is located at the end of the contact interface where the boundary condition changes from contact to no-contact. The singularity at this point is equivalent to a singularity at the end of a distributed load. Only for the case that the load value and its first derivative is zero at the end of the loading area, the singularity vanishes. As this is generally not the case for contact problems, these singularities lead to convergence rates of algebraic type for the h-, p-, and the rp-version of the FEM. The hp-version instead is able to isolate this singularity, allowing for exponential rates of convergence.