This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness-to-weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine-scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so-called inexact FETI-DP method where the local, cell-wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell-wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro-scale data, thereby making our method also multiscale and matrix-free. The solver is tested against various 2D and 3D analyses. It shows major gains with respect to black-box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes.

翻译：本文针对标准数值方法模拟超构材料所需的巨大计算资源问题展开研究。这类多尺度异质点阵结构随着增材制造技术的进步而备受关注，其具有优异的刚度重量比等特性。我们开发了一种专用高性能计算求解器，利用底层问题的特殊性质，大幅降低点阵结构全精细化分析的计算成本（内存与时间）。本方法利用自然域分解为单胞的特性，更关键的是利用了单胞间几何与力学相似性。该求解器采用非精确FETI-DP方法，其中局部单胞算子及解通过降阶建模技术进行近似。不同于独立处理每个单胞，我们仅需求解少量主要局部问题，并利用对应的主单胞算子近似所有其他单胞。该方法具有可扩展性，显著减少了局部矩阵分解次数。该求解器应用于样条组合构建点阵结构的等几何分析，可利用宏观尺度数据计算降阶基，使方法兼具多尺度特性且无需显式组装矩阵。通过多种二维与三维算例验证，相比黑箱求解器，本方法可大幅节省计算资源：在普通计算机上数分钟内即可求解数百万自由度规模的问题。