This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations at cosmological scale, and shape/topology optimization. The algorithm decomposes the simulated object into a set of convex cells called a Laguerre diagram, parameterized by the position of $N$ points in 3D and $N$ additional parameters that control the volumes of the cells. These parameters are found as the (unique) solution of a convex optimization problem -- semi-discrete Monge-Amp\`ere equation -- stemming from optimal transport theory. In this article, this setting is extended to objects with free boundaries and arbitrary topology, evolving in a domain of arbitrary shape, by solving a partial optimal transport problem. The resulting Lagrangian scheme makes it possible to accurately control the volume of the object, while precisely tracking interfaces, interactions, collisions, and topology changes.

中文翻译：

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具有自由边界的等体积拉格朗日网格的部分最优传输

本文介绍了动态网格的表示，适用于一些需要控制具有自由边界的物体体积的数值模拟，例如不可压缩流体模拟、一些宇宙尺度的天体物理模拟以及形状/拓扑优化。该算法将模拟对象分解为一组称为拉盖尔图的凸单元，由 3D 中 $N$ 点的位置和控制单元体积的 $N$ 附加参数进行参数化。这些参数被发现是凸优化问题的（唯一）解——半离散 Monge-Amp\`ere 方程——源于最优传输理论。在本文中，此设置扩展到具有自由边界和任意拓扑结构的对象，在任意形状的域中演化，通过解决部分最优运输问题。由此产生的拉格朗日方案可以精确控制对象的体积，同时精确跟踪界面、交互、碰撞和拓扑变化。