Numerical analysis of a constrained strain energy minimization problem

Technical Summary: Numerical Analysis of a Constrained Strain Energy Minimization Problem

Problem Statement
The paper addresses a specific inverse problem arising in the context of evolving surfaces implicitly represented as the zero level set of a time-dependent function $\phi(x, t)$. While the normal velocity of the surface is uniquely determined by the evolution of $\phi$, the tangential velocity component remains undetermined. This lack of tangential information means implicit surfaces do not inherently encode the trajectories of individual points on the evolving surface.

To resolve this ambiguity, the authors consider a method proposed in prior literature (specifically [12]) which seeks a "near-isometric" velocity field, also known as an approximate Killing vector field. This is achieved by minimizing a strain energy functional subject to the constraint that the velocity field's normal component matches the known normal velocity derived from $\phi$. Mathematically, for a fixed time $t$, the problem is formulated as:
$$v_T = \arg\min_{w_T} \int_{\Omega} E(u_N + w_T) : E(u_N + w_T) \, dx$$
where $u_N$ is the normal velocity, $w_T$ is the tangential component, and $E(v) = \nabla v + \nabla v^T$ is the strain tensor. The vector fields $w_T$ must be tangential to the level sets. This leads to a constrained minimization problem which is reformulated as a saddle point problem using a Lagrange multiplier.

Methodology and Mathematical Framework
The paper provides a rigorous numerical analysis of this saddle point problem and its finite element discretization.

1. Functional Setting: The problem is formulated in natural Sobolev spaces ($H^1(\Omega)$). The authors define the strain energy bilinear form $a(\cdot, \cdot)$ and introduce a constraint space $V^0_{\hat{z}}$ where the velocity is orthogonal to a given direction field $\hat{z}$ (which corresponds to the normal vector $n$ in the level set context).
2. Kernel Analysis: A critical technical issue is the non-uniqueness of the solution due to rigid motions (the kernel of the strain energy). The authors define a specific kernel subspace $K_{\hat{z}}$ (rigid motions orthogonal to $\hat{z}$) and factor this out to ensure a unique solution. They prove that in "generic" cases (e.g., when level sets are not straight lines or circles in 2D, or planar/rotationally symmetric in 3D), this kernel is trivial.
3. Saddle Point Reformulation: The constrained minimization is reformulated as an unconstrained saddle point problem. The authors introduce a Lagrange multiplier $\lambda$ to enforce the tangential constraint. They prove the well-posedness of this continuous problem, establishing the inf-sup (Ladyzhenskaya-Babuška-Brezzi) stability condition.
4. Finite Element Discretization: The authors propose a discretization using standard $H^1$-conforming finite element spaces with polynomials of degree $k$ for both the velocity and the Lagrange multiplier. They address the discrete kernel and prove discrete inf-sup stability for sufficiently small mesh sizes $h$.
5. Error Analysis: The paper derives optimal error estimates for the finite element approximation. Specifically, they prove that the error in the $H^1$-norm for the velocity and the Lagrange multiplier converges at the optimal order $O(h^k)$, provided the solution has sufficient regularity.

Key Results and Numerical Experiments
The theoretical findings are validated through several numerical experiments implemented using the Netgen/NGSolve library:

• Synthetic Example: An artificial problem with a known analytical solution confirms the theoretical convergence rates of $O(h^k)$ for the velocity error in the $H^1$-norm.
• Deforming Ellipse: The method is applied to an ellipse deforming into a biconcave shape.
  • In a regular domain without critical points of the level set function, optimal convergence rates are observed.
  • In a domain with reentrant corners, the convergence rate for the Lagrange multiplier degrades (observed as $O(h^{0.7})$ instead of $O(h^k)$), attributed to a loss of regularity in the solution near the corners.
  • In a domain containing a critical point of the level set function (where $\nabla \phi = 0$), the numerical solution exhibits singular behavior, confirming the necessity of the assumption that $\|\nabla \phi\|$ is bounded away from zero.
• Rigid Motion Recovery: For a level set transport corresponding to a rigid motion, the method successfully recovers the exact rigid motion velocity field (up to machine precision), provided the quadrature rules are sufficiently accurate to handle the non-polynomial nature of the normal vector field.
• Particle Trajectories: The paper demonstrates the practical application of the computed velocity field to track particle trajectories on a 3D bending capsule. Comparing the near-isometric velocity field against a purely normal velocity field, the results show that the near-isometric approach maintains approximately constant distances between particles, whereas the normal-only approach leads to significant particle accumulation and density variations.

Significance and Claims
The authors claim that this work provides the first detailed well-posedness analysis and error analysis for this specific constrained strain energy minimization problem and its saddle point reformulation. While the approach was previously introduced in computer graphics literature, this paper establishes the mathematical foundation for its stability and convergence.

The significance lies in:

1. Rigorous Justification: Proving that the saddle point formulation is well-posed in natural Sobolev spaces and that the discrete finite element method is stable.
2. Optimal Convergence: Deriving and verifying optimal error bounds for standard finite element spaces.
3. Handling Rigid Motions: Explicitly addressing the kernel of the strain energy operator and showing how to factor it out to ensure uniqueness.
4. Practical Utility: Demonstrating that the method effectively generates velocity fields that preserve local geometry (near-isometry), which is crucial for applications like computer graphics and shape registration where maintaining the coherence of point trajectories is essential.

The paper concludes that the method is robust for generic level set evolutions but highlights that the presence of critical points in the level set function or geometric singularities (like reentrant corners) can degrade the regularity of the solution and the convergence rates.