A regularized penalty-multiplier method for approximating cavitation solutions with prescribed cavity volume size

Let $\Omega\in\mathbb{R}^n$ be the region occupied by a body and let $\mathbf{x}_0$ be a flaw point in $\Omega$. Let $E(\cdot)$ be an energy functional (defined on some appropriate admissible set of deformations of $\Omega$). For $V>0$ fixed, we let $\mathbf{u}_V$ be a minimizer of $E(\cdot)$ among the set of deformations constrained to form a hole of volume $V$ at $\mathbf{x}_0$. In this paper we describe a regularized penalty--multiplier method and its convergence properties for the computation of both $\mathbf{u}_V$ and $E(\mathbf{u}_V)$. In particular, we show that as the regularization parameter goes to zero, the regularized constrained minimizers converge weakly in $W^{1,p}(\Omega\setminus\overline{\mathcal{B}_{\delta}(\mathbf{x}_0)})$ to $\mathbf{u}_{V}$ for any $\delta>0$. We describe as well the main features of a numerical scheme for approximating $\mathbf{u}_V$ and $E(\mathbf{u}_V)$ and give a numerical example for the case of a stored energy for an elastic fluid.