Optimal mapping is one of the longest-standing problems in computational mathematics. It is natural to measure the relative curve length error under map to assess its quality. The maximum of such error is called the quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We present a physics-based quasi-isometric stiffening (QIS) algorithm for the max-norm minimization of hyperelastic distortion.

QIS perfectly equidistributes distortion over the entire domain for the ground truth test (unit hemisphere flattening) and, when it is not possible, tends to create zones where all cells have the same distortion. Such zones correspond to fragments of elastic material that became rigid under stiffening, reaching the deformation limit. As such, maps built by QIS are related to the de Boor equidistribution principle, which asks for an integral of a certain error indicator function to be the same over each mesh cell.

Under certain assumptions on the minimization toolbox, we prove that our method can build, in a finite number of steps, a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum. We performed extensive testing: on more than 10,000 domains QIS was reliably better than the competing methods. In summary, we reliably build 2D and 3D mesh deformations with the smallest known distortion estimates for very stiff problems.

最优映射是计算数学中最古老的问题之一。很自然地测量地图下的相对曲线长度误差来评估其质量。这种误差的最大值称为拟等距常数，其最小化是一个重要的最大范数优化问题。我们提出了一种基于物理的准等距硬化（QIS）算法，用于超弹性变形的最大范数最小化。

QIS 在地面实况测试（单位半球展平）的整个域上完美地均匀分布失真，并且当不可能时，往往会创建所有单元都具有相同失真的区域。这些区域对应于弹性材料的碎片，在硬化下变得坚硬，达到变形极限。因此，QIS 构建的地图与 de Boor 等分布原理相关，该原理要求某个误差指示函数的积分在每个网格单元上都相同。

在最小化工具箱的某些假设下，我们证明我们的方法可以在有限数量的步骤中构建最大变形任意接近（未知）最小值的变形。我们进行了广泛的测试：在 10,000 多个领域中，QIS 确实优于竞争方法。总之，我们针对非常僵硬的问题，使用最小的已知失真估计可靠地构建 2D 和 3D 网格变形。