Numerical analysis of a constrained strain energy minimization problem

We consider a setting in which an evolving surface is implicitly characterized as the zero level of a level set function. Such an implicit surface does not encode any information about the path of a single point on the evolving surface. In the literature different approaches for determining a velocity that induces corresponding paths of points on the surface have been proposed. One of these is based on minimization of the strain energy functional. This then leads to a constrained minimization problem, which has a corresponding equivalent formulation as a saddle point problem. The main topic of this paper is a detailed analysis of this saddle point problem and of a finite element discretization of this problem. We derive well-posedness results for the continuous and discrete problems and optimal error estimates for a finite element discretization that uses standard $H^1$-conforming finite element spaces.

💡 Research Summary

The paper addresses a fundamental ambiguity in level‑set based surface evolution: while the normal component of the velocity field is uniquely determined by the level‑set equation, the tangential component remains under‑determined, leading to non‑unique particle trajectories on the evolving surface. To resolve this, the authors adopt the “approximate Killing vector field” approach introduced in earlier graphics literature, which selects the tangential velocity by minimizing a strain‑energy functional.

The continuous problem is formulated as follows. Given a known normal velocity $z = -\partial_t\phi / |\nabla\phi|, n$, the total velocity is written $u = z + v$, where $v$ must be pointwise orthogonal to the unit vector $\hat z$ (typically $\hat z = n$). The strain energy is defined by
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