### Face-to-Face Mortar Contact

This is a face-to-face contact formulation using extra Lagrange multipliers to model the contact stresses. It can be used for hard contact (infinite stress at the slightest penetration) or soft contact (gradually increasing stress the larger the penetration as in materials with a definite surface roughness). Due to the Lagrange multipliers the stress-penetration relationship satisfied in a weak sense. This is different from the face-to-face penalty method, in which the knowledge of the penetration uniquely leads to the contact stresses. Due to this property the convergence of the mortar method is somewhat better than in the face-to-face penalty method, i.e. less iterations are needed. However, the cost of one iteration is higher. For details the reader is referred to [82]-[85].

The implementation in CalculiX uses dual basis functions for the Lagrange multiplier. Dual basis functions are in a weak sense orthogonal to the standard basis functions used for the displacements. Due to the use of dual basis functions the Lagrange multiplier degrees of freedom can be easily eliminated from the resulting equation system and therefore the number of unknowns in the system is in each iteration not larger than without contact. Because the negative parts of the standard basis functions for quadratic elements can cause problems, several options to circumvent these problems have been implemented. Right now, the user can choose between TYPE=MORTAR, TYPE=LINMORTAR and TYPE=PGLINMORTAR on the *CONTACT PAIR card. For TYPE=MORTAR the standard dual basis functions are used for the Lagrange multiplier. For TYPE=LINMORTAR linear dual basis functions are used, i.e. the Lagrange multiplier at the midnodes (if any) is not taken into account. For linear elements MORTAR and LINMORTAR coincide. In case of TYPE=PGLINMORTAR the variation of the Lagrange multiplier is done using linear standard basis functions (PG stands for Petrov-Galerkin). The following rules apply when using Mortar contact:

• The mortar method is only available for the *STATIC procedure. Consequently, it can not be used for dynamic calculations, heat transfer calculations or (un)coupled temperature-displacement calculations, to name a few.
• It is advised to use the mortar method for contact between genuine 3-dimensional elements only. Usage for contact in between 1-d or 2-d elements will cause problems. In general, the mortar method is not well suited if the contact areas are too much constrained by extra multiple point constraints.
• The mortar method cannot be combined with the penalty method in one input file. Also a single mortar method (MORTAR or LINMORTAR or PGLINMORTAR) has to be choosen for all contact pairs in one input file.
• Using the *CYCLIC SYMMETRY MODEL option, one has to make sure that a one-to-one connection is made if hte slave surface touches the cyclic symmetry boundary. If non-matching meshes are used, one has to make sure that the contact surfaces touching the cyclic symmetry boundary are removed from the slave surface definition.
• One must not apply extra multiple point constraints to edge nodes on the slave surface. Please apply extra mounting MPC's only to corner nodes on the slave surface.
• Define different contact pairs for different contact zones (contact search algorithm is faster)
• Define contact surfaces only as large as needed (contact search algorithm is faster)
• One must not use the same contact surface in more than one contact definition
• Make sure that the contact surfaces do not touch pretension sections
• Make sure that there is not gap between the bodies for force driven quasi-static calculations (may lead to huge accelerations since no mass is defined and consequently no contact is found)
• Make sure that you choose a small first increment in the step if you expect large relative displacements in tangential direction. A minimum of four increments is recommeded. Recall that the direction of the normal and tangential directions and the surface segmentation is only performed once per increment.
• Shrink is always active in CalculiX, i.e. overlaps are resolved increment-wise across the step.
• Sometimes the adpative time stepping using mortar contact is too senstive. Try *STEP,DIRECT in that case.