# Face offsetting

**Repository Path**: li--ze/face-offsetting

## Basic Information

- **Project Name**: Face offsetting
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- **Created**: 2025-03-15
- **Last Updated**: 2025-03-27

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## README

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# Three-Dimensional Surface Evolution and Mesh Deformation for Aircraft Icing Applications

## Ⅲ. Background: Surface Evolution(表面演化)

One of the challenges associated with evolving a faceted, discrete surface representation is that the normal at a node is not unique.

一个node的法向量不是唯一的

This is caused by the discontinuous nature of the discrete representation of the surface.

这是由曲面的离散表示法的不连续性引起的。

 One possible solution is to define a displacement direction at each node, based on the normals in the adjacent faces, and displace the surface a prescribed distance in this direction at each node.

一种可能的解决方案是根据相邻facet的法向量在每个node上定义一个位移方向，并在每个节点上沿这个方向位移一个规定的距离。

However, there are numerous challenges associated with this approach, not the least of which is **conservation of volume**.

然而，与这种方法相关的是许多挑战，其中最重要的是**体积守恒**。

 however, this is not the case in three dimensions, in which any two of the nonparallel offset planes intersect in a line while any three nonparallel planes intersect at a point.

在三维中，任意两个非平行偏移平面以一条直线相交，而任意三个非平行平面相交在一点上

 In general, the intersection of four or more planes is not defined in three dimensions.

一般来说，四个或更多平面的交点不在三维中定义。

Except in special cases, the number of faces that shares a given node in a typical triangular surface mesh is usually more than three;

除特殊情况外，在一个典型的三角形曲面网格中共享一个给定节点的面数通常大于3个；

consequently, the position of the node is overspecified. This results in an ambiguity in how the nodal positions are defined in the new surface.

因此，节点的位置被过度指定了。这导致了如何在新表面中定义节点位置的模糊性。

The approach we have chosen has shown much promise for evolving a surface mesh while conserving volume

我们所选择的方法在保持体积的同时演化表面网格方面显示出了很大的前景

Since this is the basis of our surface evolution algorithm, we now describe Jiao’s algorithm in some detail.

由于这是我们的表面演化算法的基础，我们现在描述Jiao算法的一些细节。

 Jiao employed a singular value decomposition to solve a least-square problem and then applied an eigenvalue/eigenvector analysis at each node to resolve its normal motion, which generated the surface geometry, and its tangential motion, which could be used to maintain surface mesh quality.

Jiao采用奇异值分解来求解最小二乘问题，然后在每个节点上应用特征值/特征向量分析来求解其**法向运动，用于生成曲面几何**，其**切向运动，可用于保持曲面网格质量。**

step 1,face velocity、face normal 计算offset distance for each face

step 2,reconstruct the vertices(重建顶点)
$$
Nx=a
$$