For numerical simulation of elastic structures, data-driven computational approaches attempt to use a data set of material responses, without resorting to conventional modeling of the material constitutive equation. In a material data set in the stress–strain space, the data points are considered to lie on or near a low-dimensional manifold, rather distribute ubiquitously in the space. This paper presents a kernel method for extracting this manifold. We formulate a regularized least-squares problem for learning a manifold, and show that its optimal solution corresponds to an eigenvector of a real symmetric matrix. Therefore, the method requires only simple computational task, and is easy to implement. We also give a description how to use the obtained solution in static equilibrium analysis of an elastic structure. Numerical experiments on two-dimensional continua are performed to demonstrate effectiveness and robustness of the proposed method.

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一种学习数据驱动计算弹性本构关系的核方法

对于弹性结构的数值模拟，数据驱动的计算方法尝试使用材料响应的数据集，而不求助于材料本构方程的传统建模。在应力-应变空间中的材料数据集中，数据点被认为位于低维流形上或附近，而不是普遍分布在空间中。本文提出了一种用于提取这种流形的核方法。我们制定了一个用于学习流形的正则化最小二乘问题，并表明其最优解对应于实对称矩阵的特征向量。因此，该方法只需要简单的计算任务，并且易于实现。我们还描述了如何在弹性结构的静态平衡分析中使用所获得的解。