多孔狀結構[1]在自然界中十分常見，時常被應用於不同的仿生材料設計。其特殊的幾何特性往往賦予它們優異的力學性質，在工程應用與材料設計領域亦十分具有優勢，也被稱作為「多孔力學超材料」(Cellular Mechanical Metamaterials, CMMs)。傳統上，多孔力學超材料的機械性質可以透過不同數值方法計算，例如有限元素法(Finite Element Method, FEM) 。然而多孔力學超材料通常具有複雜幾何外觀，導致直接數值模擬(Direct Numerical Simulation, DNS)需要建立複雜網格而消耗可觀的計算資源。此外微觀多孔結構與其對應的力學表現具有複雜的關聯性，增加材料結構設計與優化之困難度。在此篇研究裡，我們發展了一個基於機器學習的有限元素法框架用以有效率地模擬多孔力學超材料。首先，多孔力學超材料被視為特徵體積單元(Representative Volume Element, RVE)的集成；接著，透過有限元素法計算不同形變下特徵體積單元的反應，可以獲得離線類神經網路(Artificial Neural Network, ANN)的訓練數據集。訓練過後的類神經網路將會蘊含特徵體積單元的力學性質，可被視為一均質化的元素，並且可被進一步應用於巨觀結構的有限元素模擬[2]。這樣的類神經網路在不同的邊界條件、初始形狀與材料參數中扮演了能取代此超材料本徵模型的角色。由於生成訓練數據集僅需要單一特徵體積單元進行建模，因此大幅地降低了計算成本。為了確保類神經網路與實際材料性質的擬合水準，需要在訓練集中加入足夠的形變資訊。透過在訓練集中增加微觀RVE相對應的特徵，我們可以在巨觀模擬中發現更多複雜的運動學模態。最後，我們也將透過一些實例演練去驗證此框架的有效性與穩健性。此類透過數據驅動並且結合機器學習的數值方法將能為結構設計、結構模擬與優化提供新的研究途徑。

[1] Gibson, L. J. (2003). Cellular solids. Mrs Bulletin, 28(4), 270-274.
[2] Xue, T., Beatson, A., Chiaramonte, M., Roeder, G., Ash, J. T., Menguc, Y., Adriaenssens, S., Adams, R. P. and Mao, S. (2020). A data-driven computational scheme for the nonlinear mechanical properties of cellular mechanical metamaterials under large deformation. Soft matter, 16(32), 7524-7534.

The consistency condition requires the stress locates on the yield stress during a material behaves the plastic deformation. Therefore, the capability of numerical integrations of elastoplastic models to keep the stress point on the yield surface during the plastic phase strongly influences on the efficiency, accuracy and convergence of the elastoplastic model solution. In this study, we investigate the consistency error, which measures the ability of an integration updating the stress on the yield surface when material undergoes plastic state, of a return-free integration for an anisotropic material. Taking into account uniaxial and biaxial paths in 3D stress space, we construct consistency errors of the return-free integration as well as do comparisons with results of the well-known Runge-Kutta 4th order integration. The error analysis demonstrates an acceptable accuracy of the return-free integration.

近年來，台灣的製造業已從單純代工逐漸轉變為高附加價值導向的研發與製造。以製鞋業為例，許多國際大廠的研發中心已經陸續在台灣深耕，其中如何應用仿生微結構與3D列印，設計具輕量化、高吸能、高強度等優異性能的產品則是目前研發相當重要的議題。這項研究主要集中在三重週期最小表面（TPMS）結構上。我們透過3D列印將各種型態的微結構材料列印並進行壓縮試驗，找到標準曲線得到簡化模型。經比較後Gyroid擁有較突出的機械性質，因此選擇Gyroid作為主要目標微結構。另外，由於軟硬材料交錯的生物材料具有高度的多樣性，我們則通過選擇兩種不同強度的Gyroid佈置軟硬材料來設計抗衝擊部件，並以ABAQUS模擬確認其對應的機械性質，最後透過增強式學習（Reinforcement Learning）設計相關產品。研究結果發現多功能微結構具有出色的性能，同時具有重量輕、吸能性好和強度高等特點。我們進一步使用Satra TM-142 標準測試模型。測試表明，我們的設計將保留應變能 47%，反作用力降低 43%，應力降低 86%。本研究建構的設計架構可進一步延伸至其他相關微結構設計，藉由標準曲線我們取得同型態不同規格的微結構機械性質，並針對不同性質需求進行設計。未來可將此架構應用於航空航天、軍事國防、防彈塗層和材料設計。

牙釘與其螺紋的幾何形貌對牙釘植入後傷口的骨癒合效率與骨整合有非常劇烈的影響，因此牙釘形貌最佳化是近年臨床醫學最重要的研究項目之一。傳統上常採用機械調節理論(mechano-regulation method)與有限元素法對牙釘進行最佳化，但其需耗費大量時間進行模擬與運算，因此產生了可觀的時間成本。有鑑於此，本團隊結合深度神經網路與基因演算法研發了一套牙釘的拓樸最佳化模型，此模型藉由基因演算法迭代出具特殊形貌的牙釘與螺紋，使其在兩項骨整合效率指標:骨面積(bone area, BA)和骨頭與牙釘接觸面積(bone-implant contact, BIC)上都能有優於ITI商用牙釘的表現，並大幅減少計算時間，取代機械調節理論模型。
本模型結合深度神經網路與基因演算法。在產生深度神經網路的資料集部分，首先利用已經動物實驗驗證的機械調節理論並以有限元素法軟體模擬35天內受壓下65根形貌相異之牙釘其周圍骨細胞元素產生的應變值及體液流速值，並以此計算出細胞刺激因子S。細胞刺激因子S會使骨細胞分化成纖維組織、結締組織、未成熟骨與成熟骨等不同種類的細胞，亦或是產生骨吸收。這些65根形貌相異之牙釘對應的35天骨癒合過程包含細胞分化結果分布圖，即為訓練深度神經網路的訓練資料集。訓練完的深度神經網路可以預測35天中不同形貌牙釘其周圍不同材料性質(陽氏係數、普松比、滲透率、幹細胞濃度)細胞元素分化的結果，取代有限元素法模型，大幅減少計算時間。
基因演算法則是運用在牙釘幾何形貌設計最佳化。本團隊以二維的牙釘為主體，將螺紋間隙(healing chambers)區域進行拓樸最佳化。目標函數被設定為BA值與BIC值總和，各種可能的牙釘的螺紋形貌設計被視為不同的染色體(chromosome)並在每次迭代經歷三種行為:互換 (crossover)、突變 (mutation)和篩選(selection)後改變牙釘幾何形貌使目標函數增加。目標函數會在多次的迭代後收斂並達到最大值，代表最佳化完成。最佳化完成的牙釘形貌設計會藉由深度神經網路預測細胞分化結果與計算骨整合效率，並和ITI商用牙釘進行比較。
藉由本模型所計算模擬的最佳化牙釘形貌設計，其BA值與BIC值總和可到1.9611(最大值為2)，骨整合效率遠遠高於ITI商用牙釘。而在深度神經網路取代有限元素法的部分，可將原先將近4小時的模擬縮短成2秒，大幅節省時間成本。
此模型成功地整合深度神經網路與基因演算法，節省有限元素法模擬所耗費的大量時間與計算成本，並證明利用深度神經網路取代傳統耗時耗力的演算法(如有限元素法)以增進效率這樣的觀念是可行的。在醫學上，本團隊提出的數值化模型能以相當高的效率產生最佳化牙釘形貌設計，大幅降低骨吸收。而本模型也可以藉由改進目標函數與使用更好的骨整合效率計算模型，進一步提升效能，最終使牙釘具有更好的臨床表現
關鍵詞：牙釘；拓樸最佳化；基因演算法；有限元素法；機械調節理論；深度神經網路

The elastic constants of unidirectional fiber composites with different microstructural images and volume fractions were predicted using machine learning. Both Young’s moduli and shear moduli of unidirectional composites were taken into account in the study. A repeating unit cell (RUC) with periodic boundary condition was employed to represent the complex microstructures of the unidirectional composites. The elastic constants of the RUC were evaluated using high-fidelity generalized method of cells (HFGMC) micromechanical analysis. In HFGMC model, the RUC was divided into many subcells each of which denotes either fiber phase or matrix phase. As a result, the microstructure of the unidirectional composites can be illustrated by the subcells within the RUC. With the HFGMC, the Datasets relating the microstructural images of the fiber composites to their corresponding elastic constants were generated and then used to train a convolutional neural network (CNN) model. Basically, 90% of the datasets were utilized to train the model and the remaining datasets were used for validation. The training process for the CNN model is illustrated in the Figure. To further examine the accuracy of the trained CNN model, the properties of the unidirectional composites with fiber volume fractions of 15% and 65% were modeled using HFGMC, and the results were compared with the CNN predictions. The differences were less than 3%, indicating that the machine learning network can predict the elastic constants of unidirectional composites with accuracy.

Natural materials constantly inspire us to explore and exploit materials design space for engineering applications. Many extraordinary properties from natural materials are often a direct consequence of their inherent microstructures. However, building microstructures with desired properties from scratch is not an easy task, especially to achieve those properties in conflict, for example, lightweight and strength, strength and toughness, etc. For mission-critical engineering applications, materials with high strength and toughness are of great desire. The design space inspired from nacres is of great interest to be explored. Microstructures from nacres have an inherent brick-and-mortar pattern using mostly stiff materials with a small part of soft materials to achieve high toughness and strength. In this work, we apply reinforcement learning to search the optimal microstructure with the best strength and toughness properties in the nacre design space. The deep Q-learning (DQN), a model-free reinforcement learning algorithm is applied to learn the value of an action in a particular state. We first build up a two-parameter brick-and-mortar structure environment, where the DQN agent learns from manipulating only two parameters. Next, we make our DQN agent to explore beyond a pure brick-and-mortar design space. The agent can freely place soft or stiff materials in each of the nine blocks of the unit cell. The parameters tunable in this environment are increased to 13. We also implement a Fast Fourier Transform based (FFT-based) homogenization method with linear elasticity and a non-local damage theory to calculate the ultimate tensile strength and toughness of the microstructure proposed by the DQN agent. The results suggest that our model is capable of finding brick-and-mortar structures with desired high strength and toughness. Also, for the environment beyond brick-and-mortar, our model is able to find an innovative design with desired properties. In conclusion, we expect this method can be applied not only in a small design space, but also in a large combinatorial design space, finding promising structures with desired properties possible to be used for engineering applications.

Metamaterials with particularly designed microstructure may exhibit unconventional mechanical properties, such as chiral effects, auxeticity or negative thermal expansion coefficient (NTEC). Auxetic materials exhibit negative Poisson’s ratio. Due to the internal rotational degrees of freedom at each material point in chiral materials, uniaxial-torsion or uniaxial bending deformation couplings may occur, as predicted by the non-centrosymmetric Cosserat elasticity or strain gradient elasticity. Such couplings are of great importance for novel sensor technologies. In this work, deep neural networks (DNN) are developed to learn the responses of metamaterials with randomly distributed voids or chiral microstructure, as shown in the figure below. Data, which are the effective properties of the metamaterials, can be from finite element calculations or experiments. The well-trained DNN is capable of efficiently predict the mechanical properties of the architected metamaterials, as oppose to high computational cost from finite element analysis. DNN-generated chiral microstructures may give rise to auxeticity, NTEC and coupling effects. The deformation-mode coupling may be of particular importance for designing novel sensors or actuators. Furthermore, experimental specimens are manufactured from 3D printing techniques to cross-validate the DNN results.