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\({\text {FE}}^r\) method with surrogate localization model for hyperelastic composite materials
Advanced Modeling and Simulation in Engineering Sciences ( IF 2.0 ) Pub Date : 2020-10-06 , DOI: 10.1186/s40323-020-00175-0 Ryo Hatano , Seishiro Matsubara , Shuji Moriguchi , Kenjiro Terada , Julien Yvonnet
Advanced Modeling and Simulation in Engineering Sciences ( IF 2.0 ) Pub Date : 2020-10-06 , DOI: 10.1186/s40323-020-00175-0 Ryo Hatano , Seishiro Matsubara , Shuji Moriguchi , Kenjiro Terada , Julien Yvonnet
This study presents a method for constructing a surrogate localization model for a periodic microstructure, or equivalently, a unit cell, to efficiently perform micro-macro coupled analyses of hyperelastic composite materials. The offline process in this approach is to make a response data matrix that stores the microscopic stress distributions in response to various patterns of macroscopic deformation gradients, which is followed by the proper orthogonal decomposition (POD) of the matrix to construct a reduced order model (ROM) of the microscopic analysis (localization) with properly extracted POD bases. Then, response surfaces of the POD coefficients are constructed so that the ROM can be continuous with respect to the input datum, namely, the macroscopic deformation gradient. The novel contributions of this study are the application of the L2 regularization to the interpolation approximations of the POD coefficients by use of radial basis functions (RBFs) to make the response surfaces continuous and the combined use of the cross-validation and the Bayesian optimization to search for the optimal set of parameters in both the RBFs and L2regularization formula. The resulting model can be an alternative to microscopic finite element (FE) analyses in the conventional $${\text {FE}}^2$$ method and realizes $${\text {FE}}^r$$ with $$1
中文翻译：
超弹性复合材料的具有替代定位模型的\（{\ text {FE}} ^ r \）方法
这项研究提出了一种方法，用于构造周期性微结构（或等效地，晶胞）的替代物定位模型，以有效地进行超弹性复合材料的微宏耦合分析。此方法的离线过程是制作一个响应数据矩阵，该响应数据矩阵存储响应于宏观变形梯度的各种模式的微观应力分布，然后进行矩阵的适当正交分解（POD）以构造降阶模型（ ROM）的显微镜分析（定位），并正确提取POD碱基。然后，构造POD系数的响应面，使得ROM可以相对于输入数据（即，宏观变形梯度）连续。这项研究的新颖贡献是通过使用径向基函数（RBF）将L2正则化应用于POD系数的插值近似，以使响应面连续，以及交叉验证和贝叶斯优化的组合使用。在RBF和L2正则化公式中搜索最佳参数集。所得模型可以替代常规$$ {\ text {FE}} ^ 2 $$方法中的微观有限元（FE）分析，并通过$$ 1实现$$ {\ text {FE}} ^ r $$
更新日期：2020-10-07
中文翻译：
超弹性复合材料的具有替代定位模型的\（{\ text {FE}} ^ r \）方法
这项研究提出了一种方法，用于构造周期性微结构（或等效地，晶胞）的替代物定位模型，以有效地进行超弹性复合材料的微宏耦合分析。此方法的离线过程是制作一个响应数据矩阵，该响应数据矩阵存储响应于宏观变形梯度的各种模式的微观应力分布，然后进行矩阵的适当正交分解（POD）以构造降阶模型（ ROM）的显微镜分析（定位），并正确提取POD碱基。然后，构造POD系数的响应面，使得ROM可以相对于输入数据（即，宏观变形梯度）连续。这项研究的新颖贡献是通过使用径向基函数（RBF）将L2正则化应用于POD系数的插值近似，以使响应面连续，以及交叉验证和贝叶斯优化的组合使用。在RBF和L2正则化公式中搜索最佳参数集。所得模型可以替代常规$$ {\ text {FE}} ^ 2 $$方法中的微观有限元（FE）分析，并通过$$ 1实现$$ {\ text {FE}} ^ r $$
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