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Treatment of Inequality Constraints

To impose the displacement compatibility condition along the contact area, additional Lagrange multiplier degrees of freedom, $\lambda$, are introduced to the system to minimize the potential function $\pi$ under the contact constraints defined by the function $g({\bf u},{\bf\dot{u}})=0$. By adding the contact conditions to the usual variational indicator, the following function is obtained

 \begin{displaymath}\Pi({\bf u},{\bf\dot{u}})=\pi({\bf u},{\bf\dot{u}})+\mbox{{\boldmath$\lambda$ } } g({\bf u},{\bf\dot{u}})

The second term of the equation may be physically interpreted as the potential of the contact forces. The values of the vector $\lambda$ are the contact forces resulting from imposing the contact conditions on the system.

The friction part of the force transmissibility condition is not enforced yet in Equation (3.5). This condition is handled separately after each iteration and the appropriate boundary conditions are set for the next iteration. This approach causes the status of the boundary conditions applied on the system to change after each iteration. As a result, the quadratic convergence of the Newton-Rapson method may be seriously affected.

A. Zeiny