Functionals that penalize bending or stretching of a surface play a key role in geometric and scientific computing, but to date have ignored a very basic requirement: in many situations, surfaces must not pass through themselves or each other. This paper develops a numerical framework for optimization of surface geometry while avoiding (self-)collision. The starting point is the tangent-point energy, which effectively pushes apart pairs of points that are close in space but distant along the surface. We develop a discretization of this energy for triangle meshes, and introduce a novel acceleration scheme based on a fractional Sobolev inner product. In contrast to similar schemes developed for curves, we avoid the complexity of building a multiresolution mesh hierarchy by decomposing our preconditioner into two ordinary Poisson equations, plus forward application of a fractional differential operator. We further accelerate this scheme via hierarchical approximation, and describe how to incorporate a variety of constraints (on area, volume, etc.). Finally, we explore how this machinery might be applied to problems in mathematical visualization, geometric modeling, and geometry processing.

中文翻译：

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排斥面

惩罚曲面弯曲或拉伸的泛函在几何和科学计算中发挥着关键作用，但迄今为止忽略了一个非常基本的要求：在许多情况下，曲面不能通过自身或彼此。本文开发了一个用于优化表面几何形状同时避免（自）碰撞的数值框架。起点是切点能量，它有效地将空间上靠近但沿表面远离的点对分开。我们为三角形网格开发了这种能量的离散化，并引入了一种基于分数 Sobolev 内积的新型加速方案。与为曲线开发的类似方案相比，我们通过将预处理器分解为两个普通泊松方程，避免了构建多分辨率网格层次结构的复杂性，加上分数微分算子的前向应用。我们通过层次逼近进一步加速了这个方案，并描述了如何结合各种约束（面积、体积等）。最后，我们探索如何将这种机制应用于数学可视化、几何建模和几何处理中的问题。