In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE2 and IGA2 approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null-space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE2 and IGA2 formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general non-linear problems.

翻译：本文为计算均匀化方法导出了一致的变分公式，并证明传统的FE²和IGA²方法只是该最一般框架的特定离散化与求解技术。这使得我们能够显著增强数值分析以及所生成代数系统的求解能力。具体而言，我们通过扩展连续系统的维度，对高维问题进行一致离散化，随后应用离散零空间矩阵移除额外维度。针对一个可获得解析解的基准问题，我们证明了所选方法在将传统FE²和IGA²公式的巨额计算成本降低至原始需求一小部分方面的优越性。最后，我们展示了在一般非线性问题求解中进一步降低计算成本的方法。