等几何分析（IgA）中基于投影的低秩组装方法
Projection-based low-rank assembly in IgA
- Technical University Chemnitz（化学nitz工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对等几何分析（IgA）中质量与刚度矩阵组装存储成本高的问题，提出基于投影的低秩方法，在TT格式下结合AMEn方法求解，适用于奇异插值系统和近奇异几何场景，性能可与全组装及插值低秩方法媲美。
AI中文摘要：
等几何分析（IgA）采用相同的样条函数表示计算域并近似解，这可实现精确的几何描述，但生成的质量矩阵和刚度矩阵的组装与存储成本较高，尤其在三维场景中。本文提出一种基于投影的低秩方法，用于组装保定向张量积B样条几何的质量张量和刚度张量。对于质量张量，我们利用几何映射雅可比行列式的多项式结构，通过单变量系数转移算子将其表示在降维样条乘积空间中；结合精确求积且无截断，得到的低秩张量是标准Galerkin质量张量的精确重构。对于刚度张量，我们将有理权重函数拆分为多项式分子（同样表示在降维样条乘积空间中）和倒数行列式项（其通常非样条函数，因此通过张量积样条空间上的L²投影近似）。两种构造均完全在张量列车（TT）格式下进行，投影系统通过交替最小能量（AMEn）方法求解，因此无需形成完整的高阶系数张量，多维积分可简化为单变量积分与收缩乘积。该方法在MATLAB中实现，使用GeoPDEs和TT-Toolbox。数值实验表明，该方法的性能可与全组装方法及基于插值的低秩方法相媲美，且适用于插值存在问题的两种场景：奇异插值系统和近奇异几何。该构造仅限于保定向张量积B样条几何，不覆盖非均匀有理B样条（NURBS）。
英文摘要：
Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS.
