Surrogate Computational Homogenization for Three-Dimensional Two-Scale Problems Based on RBF Interpolation With Decision Tree

  • Yamanaka, Yosuke (Nihon University)
  • Hirayama, Norio (Nihon University)
  • Terada, Kenjiro (Tohoku University)

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We propose a surrogate computational homogenization (SCH) method based on partitioned radial basis function (RBF) interpolation combined with a decision tree model. This approach enables machine learning-based multiscale analysis for computationally intensive three-dimensional (3D) two-scale structural problems involving heterogeneous materials. Multiscale analysis methods based on computational homogenization (CH) are now commonly used for numerical simulation of structures composed of heterogeneous materials. However, conventional methods face challenges related to computational cost and formulation. To overcome these difficulties, SCH methods that incorporate machine learning into CH have been developed in recent years. Our group has proposed a method to create SCH models that predict macroscopic stress from macroscopic strain input using RBF interpolation—an interpolation method for high-dimensional training data that creates high-fidelity functions. Although results from this method have shown good agreement with conventional approaches, a challenge remains regarding the computational cost of machine learning for fully 3D two-scale problems. Specifically, the computational cost of RBF interpolation is O(N^3 ) for N training data, and it becomes prohibitively large for 3D problems due to the significant increase in training data required to keep accuracy of an SCH model. To address this issue, we apply RBF interpolations to subdomains created by dividing the training data space using a decision tree model. The responses from each interpolant are then weighted and averaged to output the macroscopic stress. This division reduces computational cost to O(N^3/M^2 ), where M is the number of subdomains, while maintaining accuracy. The effectiveness of the proposed method is demonstrated by applying it to three-dimensional (3D) two-scale problems of structures composed of hyperelastic heterogeneous materials.