距离场隐含几何信息（嵌入）；提高搜索效率；并行

强调（额外）（高阶）：

• 时空精度
• 减少非守恒误差

调研：西工大 蔡晋生 zhengyao ? 闫超 二九基地的 谁 李书杰？not found

Mesh of background (Cartesian): \(G\)

Mesh (Unstructured) \(A\), \(B\) …

For mesh \(A\):

• 2 kinds of boundaries: Wall and Far
• Distance field: \(d_A\)
• Special definitions:
  • Beyond Wall boundaries: \(d_A<0\) or \(d_a \equiv-\infty \)
  • Beyond Far boundaries: \(d_a\equiv\infty\)
• Interpolation onto $G$:
  • $d_{G,A}$
  • Refinement:
    • In Cartesian cell $\Omega_{G,i}$ of $G$
    • If $\Omega_{G,i}$ has intersection with $\partial \Omega_{A,\text{wall}}$, the $\partial \Omega_{A,\text{wall}}$ mesh is partially stored at $\Omega_{G,i}$, and local $d_{G,A}$ is refined using more points or local bnd representation
  • Therefore, it can be guaranteed:
    • Given arbitrary point, (to lookup from $G$ or $A$) $d_{G,A}$’s determination of $d_{G,A}\leq0$, $d_{G,A}=\infty$ is identical with $d_A$

For 2 meshes $A$ and $B$:

• For each cell $\Omega_{A,i}$ in $A$:
  • Query points $p \in \Omega_{A,i}$
  • We have $d_{G,A}(p),d_{G,B}(p)$
    • If $\min{d_{G,A}(p),d_{G,B}(p)} < 0 \text{ or } \equiv -\infty$, set $p\in\Omega_{Hole}$
    • If $\min{d_{G,A}(p),d_{G,B}(p)} > 0$ and $\arg\min{d_{G,A}(p),d_{G,B}(p)} = A$, set $p\in\Omega_{A,Field}$
    • If $\min{d_{G,A}(p),d_{G,B}(p)} > 0$ and $\arg\min{d_{G,A}(p),d_{G,B}(p)} = B$, set $p\in\Omega_{A,Recv,B}$