Abstract The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method.

中文翻译：

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基于深度神经网络和等几何多网格设计方法的三向功能梯度板材料优化

摘要 本文旨在提高计算性能，以优化三向功能梯度 (FG) 板的材料分布。我们利用使用非均匀有理 B 样条 (NURBS) 基函数的优势来描述在功能梯度 (FG) 板的所有三个方向上变化的材料分布。首次提出了使用广义剪切变形理论 (GSDT) 和等几何分析 (IGA) 相结合的多方向（1D、2D 和 3D）FG 板的二维自由振动和屈曲行为。这种方法可以帮助节省大量的计算成本，同时仍然确保解决方案的准确性。通过与文献中的其他方法进行比较，证明了本方法的有效性和可靠性。所得结果与参考结果非常吻合。更重要的是，由输入-输出对组成的数据集是从分析过程中通过深度神经网络 (DNN) 训练过程的迭代随机生成的。DNN 被用作分析工具以取代有限元分析以降低计算成本。通过使用 DNN，可以从这些材料分布直接预测多向 FG 板的行为。首次使用改进的共生生物搜索（mSOS）算法找到了在各种体积分数约束下自由振动或压缩下的三向 FG 板的最佳材料分布。此外，等几何多网格设计技术也用于减少优化中的大量设计变量。