Construction of signed distance functions with an elliptic equation

Abstract

Motivated by recent progress of structural optimization problems, the paper presents a new method for constructing the distance function to the boundary of given sets of interest, which simplifies the optimization procedure. We extend the celebrated Varadhan's elliptic equation theory in 1967 by adding the source term to the equation, in which we encode the information about the set. We will also establish the rate of convergence in this new framework, which is sharp at least in the one-dimensional case.

Figures (3)

• Figure 1: The graphs of $f(x)$ (left), $-\sqrt{a} \log u_a (x)$ (right, solid line) and $k-|x|$ (right, dashed line) in the setting of Example \ref{['ex:1D']} with $h=1$, $k=\frac{2}{3}$, $\alpha=1$, $a=0.0001$ and $\zeta=2$.
• Figure 2: A typical configuration of $\Omega$ and $A =A_1 \cup A_2 \cup A_3$.
• Figure 3: A typical configuration of $\Omega$, $A =A_1 \cup A_2 \cup A_3$ and $B=B_{\sqrt{a}}(\tilde{y}).$

Theorems & Definitions (13)

• Example 1.1
• Theorem 1.2: Construction of distance function
• Remark 1.3
• Theorem 1.4: Rate of convergence
• Theorem 1.5: Construction of (local) signed distance function
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1