AI 中文总结
本研究提出张量列车（TT）格式求解等几何分析离散的大规模三维线性弹性问题，对比两种算子存储策略及对应求解器，验证其在压缩算子和解、提升模拟效率与可扩展性上的优势。
AI 中文摘要
本研究开发张量列车（TT）格式，用于求解等几何分析离散得到的大规模三维线性弹性问题。通过利用基函数的张量积结构以及依赖几何的系数场的低秩结构，将刚度算子、质量算子、力向量和位移解表示为TT格式。研究了两种求解策略：块算子格式，其中耦合弹性算子以分离的TT块存储；单算子格式，其中完整耦合系统以单个整体TT算子存储。针对块格式引入无矩阵的三场TT共轭梯度求解器，而单算子格式使用AMEn求解器。数值算例表明，与传统的稀疏全网格表示相比，算子和解均实现了显著压缩，证明基于TT的格式为大规模三维弹性模拟提供了高效且可扩展的方法。
英文摘要
This work develops tensor-train (TT) formulations for solving large-scale three-dimensional linear elasticity problems discretized by isogeometric analysis. By exploiting the tensor-product structure of the basis functions and the low-rank structure of geometry-dependent coefficient fields, the stiffness operator, mass operator, force vector, and displacement solution are represented in TT format. Two solution strategies are investigated: a block-operator formulation, in which the coupled elasticity operator is stored as separated TT blocks, and a single-operator formulation, in which the full coupled system is stored as one monolithic TT operator. A matrix-free three-field TT conjugate-gradient solver is introduced for the block formulation, while AMEn is used for the single-operator formulation. Numerical examples demonstrate substantial compression of both operators and solutions compared with conventional sparse full-grid representations, showing that TT-based formulations provide an efficient and scalable approach for large-scale three-dimensional elasticity simulations.
