Penalty Method

In the solution of constrained optimization problems, penalty methods consist in replacing the constrained optimization problem with a sequence of unconstrained optimization problems. The virtual power continues to be minimized so as to find the stationary condition, but a penalty term is added to Virtual Power Term Names, Equation 5 so as to impose the impenetrability condition:

Where,

$\varphi \left({\gamma }_N\right)=0$ if ${\gamma }_N=0$

$\varphi \left({\gamma }_N\right)>0$ if ${\gamma }_N<0$

$\rho$ is an arbitrary parameter known as the penalty parameter. The penalty function $\varphi$ is an arbitrary function of the interpenetration and its rate. It is emphasized that the weak form, including the virtual power and the penalty term Penalty Method, Equation 1 is not an inequality form. The penalty function will be defined in the description of interfaces.