The FE² method is a very flexible but computationally expensive tool for multiscale simulations. In conventional implementations, the microscopic displacements are iteratively solved for within each macroscopic iteration loop, although the macroscopic strains imposed as boundary conditions at the micro-scale only represent estimates. In order to reduce the number of expensive micro-scale iterations, the present contribution presents a monolithic FE² scheme, for which the displacements at the micro-scale and at the macro-scale are solved for in a common Newton–Raphson loop. In this case, the linear system of equations within each iteration is solved by static condensation, so that only very limited modifications to the conventional, staggered scheme are necessary. The proposed monolithic FE² algorithm is implemented into the commercial FE code Abaqus. Benchmark examples demonstrate that the monolithic scheme saves up to $\approx$ 60% of computational costs.

FE ²方法是用于多尺度仿真的非常灵活但计算量大的工具。在常规实施方式中，尽管在微观尺度上作为边界条件施加的宏观应变仅表示估计，但是对于每个宏观迭代循环内的微观位移都是迭代求解的。为了减少昂贵的微尺度迭代次数，本贡献提出了整体式FE ²方案，在一个通用的牛顿-拉夫森环中解决了微观尺度和宏观尺度的位移。在这种情况下，每次迭代中的线性方程组通过静态凝聚法求解，因此仅需对常规的交错方案进行非常有限的修改即可。所提出的整体式FE ²算法被实现为商业化的FE代码Abaqus。基准测试示例表明，整体方案最多可节省$\approx$ 计算成本的60％。