Level Set Description Research Articles

Primal–dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration

Based on a level-set description of a crack moving with a given velocity, the problem of shape perturbation of the crack is considered. Nonpenetration conditions are imposed between opposite crack surfaces which result in a constrained minimization problem describing equilibrium of a solid with the crack. We suggest a minimax formulation of the state problem thus allowing curvilinear (nonplanar) cracks for the consideration. Utilizing primal-dual methods of shape sensitivity analysis we obtain the general formula for a shape derivative of the potential energy, which describes an energy-release rate for the curvilinear cracks. The conditions sufficient to rewrite it in the form of a path-independent integral (J-integral) are derived.

INTERFACE STRUCTURE

Phase-separated fluids are analyzed from several viewpoints: local thermodynamic, in which all density correlations are ignored; mean field, which correctly treats those at medium range; and a modified Kac-Siegert form which only coarse-grains those at short range. The last form appears as an ensemble average over tail potential fields, and is treated variationally with a Gaussian ansatz. Application to a planar interface shows the anticipated surface softening as any confining field is removed, and in statistically homogeneous condensation, a form of the familiar level set description assumes relevance. Suggestions are made as to removing the Gaussian assumption.