基于HDMR神经网络逼近器的有限应变弹性静力学计算均匀化代理模型
A surrogate model for computational homogenization of elastostatics at finite strain using the HDMR-based neural network approximator
AI总结：
提出一种基于HDMR神经网络代理模型，用于有限应变弹性静力学双尺度均匀化，通过显式宏观能量密度实现离线-在线两阶段高效计算。
摘要：
我们提出了一种用于有限应变下弹性静力学双尺度计算均匀化的代理模型。宏观本构定律通过相关宏观能量密度的显式公式在数值上变得可用。该能量密度通过使用模拟高维模型表示的神经网络架构来构建。用于训练该网络的数据集是通过求解一组微观边界值问题来组装的，这些问题具有指定的宏观变形梯度（输入数据），随后检索相应的平均能量（输出数据）。因此，非线性弹性的双尺度计算过程可以分解为两个求解器，分别用于微观和宏观平衡方程，它们分两个阶段独立工作，称为离线阶段和在线阶段。采用标准有限元方法求解宏观尺度的平衡方程。对于微观问题，采用基于FFT的配置方法，与牛顿-拉夫逊迭代和共轭梯度法配合使用。特别地，我们以Lippmann-Schwinger形式求解微观平衡方程，而不借助参考介质，从而避免了在非线性区域可能要求相当严格数值稳定性条件的定点迭代。
英文摘要：
We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macro-energy density. This energy density is constructed by using a neural network architecture that mimics a high-dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary values problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two-scale computational procedure for the nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. A standard finite element method is employed to solve the equilibrium equation at the macroscale. As for mircoscopic problems, an FFT-based collocation method is applied in tandem with the Newton-Raphson iteration and the conjugate-gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium and thus avoid the fixed-point iteration that might require quite strict numerical stability condition in the nonlinear regime.
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