The solution of computational fluid dynamics problems is one of the most computationally hard tasks, especially in the case of complex geometries and turbulent flow regimes. We propose to use Tensor Train (TT) methods, which possess logarithmic complexity in problem size and have great similarities with quantum algorithms in the structure of data representation. We develop the Tensor train Finite Element Method -- TetraFEM -- and the explicit numerical scheme for the solution of the incompressible Navier-Stokes equation via Tensor Trains. We test this approach on the simulation of liquids mixing in a T-shape mixer, which, to our knowledge, was done for the first time using tensor methods in such non-trivial geometries. As expected, we achieve exponential compression in memory of all FEM matrices and demonstrate an exponential speed-up compared to the conventional FEM implementation on dense meshes. In addition, we discuss the possibility of extending this method to a quantum computer to solve more complex problems. This paper is based on work we conducted for Evonik Industries AG.

翻译：计算流体动力学问题的求解是计算难度最大的任务之一，尤其在复杂几何构型与湍流流态的情况下。我们提出采用张量列方法，该方法在问题规模上具有对数复杂度，且数据表示结构与量子算法高度相似。我们开发了张量列有限元方法——TetraFEM，并构建了基于张量列求解不可压缩Navier-Stokes方程的显式数值格式。我们在T型混合器的液体混合模拟中测试了该方法，据我们所知，这是首次将张量方法应用于此类非平凡几何构型。如预期所示，所有有限元矩阵均实现了指数级内存压缩，并且在密集网格上的计算速度相较于传统有限元方法实现了指数级加速。此外，我们探讨了将该方法拓展至量子计算机以解决更复杂问题的可能性。本文基于我们为赢创工业集团开展的研究工作。