# Interfaces

Interfaces solve the contact and impact conditions between two parts of a model.

Contact-impact problems are among the most difficult nonlinear problems to solve as they introduce discontinuities in the velocity time histories. Prior to the contact, the normal velocities of the two bodies which come into contact are not equal, while after impact the normal velocities must be consistent with the impenetrability condition. In the same way, the tangential velocities along interfaces are discontinuous when stick-slip behavior occurs in friction models. These discontinuities in time complicate the integration of governing equations and influence performance of numerical methods.

Central to the contact-impact problem is the condition of impenetrability. This condition states that bodies in contact cannot overlap or that their intersection remains empty. The difficulty with the impenetrability condition is that it cannot be expressed in terms of displacements as it is not possible to anticipate which parts of the bodies will come into contact. For this reason, it is convenient to express the impenetrability condition in a rate form at each cycle of the process. This condition can be written as:(1)
${\gamma }_{N}={v}_{N}^{A}-{v}_{N}^{B}\le 0$

on the contact surface ${\Gamma }_{C}$ common to the two bodies.

${v}_{N}^{A}$ and ${v}_{N}^{B}$ are respectively the normal velocities in the two bodies in contact. ${\gamma }_{N}$ is the rate of interpenetration.

Equation 1 simply expresses that when two bodies are in contact, they must either remain in contact and ${\gamma }_{N}=0$ , or they must separate and ${\gamma }_{N}<0$ .

On the other hand, the tractions must observe the balance of momentum across the contact interface. This requires that the sum of the tractions on the two bodies vanish:(2)
${t}_{N}^{A}+{t}_{N}^{B}=0$
Normal tractions are assumed compressive, which can be stated as:(3)
${t}_{N}={t}_{N}^{A}=-{t}_{N}^{B}<0$

Equation 1 and Equation 2 can be combined in a single equation stating that, ${t}_{N}{\gamma }_{N}=0$ . This condition simply expresses that the contact forces do not create work. If the two bodies are in contact, the interpenetration rate vanishes. On the other hand, if the two bodies are separated ${\gamma }_{N}<0$ but the surface tractions vanish. As a result, the product of the surface tractions and the interpenetration rate disappear in all cases.

The impenetrability condition is expressed as an inequality constraint, the condition:(4)
${t}_{N}{\gamma }_{N}=0$

can also be seen as the Kuhn-Tucker condition associated with the optimization problem consisting in minimizing the total energy (Virtual Power Principle, Equation 5) subject to the inequality constraint Lagrange Multiplier Method, Equation 1.

In practice, the solution to a contact problem entails in three steps:
• First, it is necessary to find for each point those points in the opposite body which will possibly come into contact. This is the geometrical recognition phase.
• The second phase is to check whether or not the bodies are in contact and, if the bodies are in contact, if they are sticking or slipping. This step makes use of the geometrical information computed in the first phase.
• The last step will be to compute a satisfactory state of contact.
The geometrical recognition phase is dependent on the type of interface. This will be discussed below in parallel with the description of interfaces. On the other hand, structural problems with contact-impact conditions lead to constrained optimization problems, in which the objective function to be minimized is the virtual power subject to the contact-impact conditions. There are conventionally two approaches to solving such mathematical programming problems:
• the Lagrange multiplier method
• the Penalty method.

Both methods are used in Radioss.