力學超材料之材料性質主要由結構而非組成所決定。其中，三度週期最小曲面（TPMS）是眾所周知的力學超材料之一，這樣的結構內充滿了扭曲的通道與複雜的幾何。傳統上，有限元素法（FEM）可以被用於模擬力學超材料，但由於存在許多缺陷和扭曲的幾何形狀，用於模擬之模型變得難以建構。在所建構之TPMS模型中容易出現過多低質量的網格，因此直接數值模擬（DNS）不但計算效率低且容易變得無效。
為了解決這個問題，本研究開發了一種基於類神經網路的有限元方法（NN-FEM）。在這個框架中，可以使用替代模型而不需要詳細的結構拓撲。類神經網路（ANN）的建立替代了傳統的解析本構模型進行數值模擬。為了離線訓練神經網路，在微觀尺度中的特徵體積單元（RVE）需要被定義，並且透過施加Dirichlet邊界條件來獲取其力學表現以建構Helmholtz自由能數據集。在替代模型中使用RVE，可以透過一個均質的網格模型來模擬復雜的TPMS結構。TPMS的參數化結構訊息也可以嵌入到ANN模型中，以解決具有獨特幾何形狀的RVE之隱性本構關係。本研究提供了諸多數值示例如懸臂梁彎曲測試，用以評估NN-FEM在參數化TPMS結構中的有效性和效率。該模型可應用於人類股骨的快速應力分析和骨質疏鬆症檢測。此方案為設計、建模和優化複雜工程材料提供了一種新穎的高效方式。

Mechanical metamaterials are artificial structures where the structure rather than the composition primarily determines their material properties. Triply periodic minimal surface (TPMS) is one of the well-known metamaterials; its volume is occupied with distorted morphology and sophisticated tunnel systems. In tradition, the finite element method (FEM) is employed to simulate the metamaterials, but plenty of defects and twist geometry pose difficulty in the model construction. Excessive low-quality mesh is found in the TPMS model, and therefore, the direct numerical simulation (DNS) becomes computationally inefficient and ineffective.
To resolve this issue, this study develops a neural network enhanced finite element method (NN-FEM) for parameterized TPMS multiscale meta-structures. A 2-D porous structure, Gyroid structure with different volume fractions, and Diamond structure with different topology constants are demonstrated in this study. In this framework, a surrogate model can be employed without detailed structural topologies, and the hypothesis constructed by the artificial neural network (ANN) replaces the traditional analytical constitutive model in the numerical simulation. In order to train the neural network offline, the representative volumetric element (RVE) is defined at a local scale, and its mechanical responses are obtained by imposing Dirichlet boundary conditions to acquire the Helmholtz free energy datasets. The usage of RVE in the surrogate model results in a homogeneous mesh to model a complicated TPMS structure. The NN-trained surrogate material model is deployed on the macroscale Galerkin finite element model for various field tests. The parameterized structural information of TPMS can also be embedded into the NN model to resolve the hidden constitutive relation of RVEs with unique geometries. Numerical examples, such as the cantilever beam bending test, are provided to benchmark the effectiveness and efficiency of NN-FEM for parameterized TPMS structures. The proposed model can be applied to femurs for rapid stress analysis and osteoporosis detection. The scheme provides a new and efficient way to design, model, and optimize complicated engineering metamaterials.

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