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

Abstract

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.

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