Spatial Eigenanalysis of 2D Deformation Energies

We present the first closed‐form, per‐element eigensystems for 2D deformation energies in the primary domain of spatial coordinates. Previous analyses have only been able to find such eigensystems in proxy coordinates such as the deformation gradient, but as the spatial positions directly correspond to global degrees of freedom, our analysis opens up avenues for novel algorithms that were not possible before. First, we propose a spectral preconditioner that approximates global near‐nullspace modes using local analytical eigenvectors and clusters global eigenvalues using a low‐rank correction. Second, we derive a unified projected Newton formulation that combines the element eigensystem with the mass and damping terms, yielding a superior eigenvalue filter. We test our preconditioner on surface parameterization problems and our projected Newton framework on dynamic simulation examples. The results show significant efficiency gains over previous methods. In particular, our physics‐aware filtering strategy reduces eigenvalue clamping by up to 30×, which in turn decreases the Newton iteration count by roughly 2×.