Adaptive estimation of Hodge star operator on simplicial surfaces

Abstract

The Hodge star operator is a fundamental component of the second-order differential operators that bridges constitutive physical laws, as a matter of fact, it plays a central role in geometry processing and physical simulation. To be admissible, the discrete Hodge operators should be regular, symmetric, sparse and positive definite. Unfortunately, the last criteria is rarely met in the literature, which leads to inconsistent physical simulation behavior. In this paper, we exploit the intrinsic relationship between Hodge operator and the physical Fourier’s constitutive laws, to construct an adaptive discrete Hodge operator by expressing the Fourier’s laws on a surface mesh. As by-product, the new discrete Hodge operator is diagonal, regular and positive definite. Various comparative examples are presented to demonstrate the performance of our approach. The results show that the proposed operator performs better compared with the standard discrete Hodge.

Appendix

For a sake of reproducibility, we present in what follows computational detail of discrete Harmonic 1-form (29). According to the Hodge theorem [27], each harmonic form $h^1$ is linked to a closed form $\omega ^1$ by

$$\begin{aligned} \omega ^1=d_0 \psi ^0+ h^1 \end{aligned}$$

(30)

where $d_0$ is the exterior derivative operator (1) mapping the 0-form (function) $\psi ^0$ to a 1-form. Since the harmonic form is co-closed, i.e., $\delta _1 h^1=0$, the 0-form $\psi ^0$ can be expressed in terms of the closed 1-form $\omega ^1$ as

$$\begin{aligned} \delta _1 d_0 \psi ^0 =\delta _1\omega ^1. \end{aligned}$$

(31)

On the simplicial surface ${\mathcal {M}}_2$, let us denote by ${\mathcal {H}}^1$ the vector of discrete harmonic form, $\varPsi ^0$ the vector of unknown discrete 0-form and ${\mathcal {W}}^1$ the vector of discrete closed 1-form. Then, by substituting the differential operators $d_0$ and $\delta _1$ by their matrix formulation (2) and (5), the two systems (31) and (30) can then be, respectively, expressed in discrete setting as

$$\begin{aligned}&D^T_{0} H_1 D _{0}\varPsi ^0 = D^T_{0} H_1 {\mathcal {W}}^1 \end{aligned}$$

(32)

$$\begin{aligned}&{\mathcal {H}}^1 = D_0 \varPsi ^0-{\mathcal {W}}^1,\; \end{aligned}$$

(33)

where the matrix factor $H^{-1}_0$ was simplified from both sides of the system (32), and the construction of 1-form harmonic basis is described in algorithm 2.

The discrete closed 1-form ${\mathcal {W}}^1$ in the input is computed by the co-reduction method implemented in the Gmsh library [34].