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, 式 5 so as to impose the impenetrability condition:(1)
δQ= Γ c ρϕ( γ N )dΓ


ϕ( γ N )=0 if γ N = 0

ϕ( γ N )>0 if γ N < 0

ρ is an arbitrary parameter known as the penalty parameter. The penalty function ϕ 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, 式 1 is not an inequality form. The penalty function will be defined in the description of interfaces.