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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 energyfunctional (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--multipliermethod and its convergence properties for the computation of both$\mathbf{u}_V$ and $E(\mathbf{u}_V)$. In particular, we show that as theregularization parameter goes to zero, the regularized constrained minimizersconverge 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 anumerical scheme for approximating $\mathbf{u}_V$ and $E(\mathbf{u}_V)$ andgive a numerical example for the case of a stored energy for an elastic fluid.

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