利用最佳化演算法於週期結構薄板以振動特性反算積層製造之正交性材料常數

Abstract

本研究旨在針對自由邊界正交性材料矩形薄板，建立一套基於共振頻率與模態振形之非破壞材料常數量測方法。透過實驗量測薄板共振頻率與模態，結合疊加法理論與最佳化演算法，反算九個獨立正交性材料常數。 實驗部分以熱熔堆疊式3D列印製作三種不同列印方向之實心與網格週期結構試片，經由喇叭激振，並使用雷射都卜勒振動儀量測共振頻率，搭配電子斑點干涉術擷取模態振形。 理論推導方面，本研究基於正交性材料關係搭配薄板假設，由面外振動的統御方程式利用疊加法滿足特定邊界條件，再透過位移函數的正交性解析特徵值問題，求得薄板之共振頻率與模態振形，並與有限元素模擬進行相互驗證。 本研究最佳化演算法以基因演算法為核心，針對其於高維多峰非線性問題中，易陷入局部最小值與過早收斂之缺點，進而結合自適應模擬退火以強化搜尋能力，雖可跳脫局部搜尋限制，但全域探索與穩定性仍不足。為克服此問題，本研究最終改以混合型基因演算法-螞蟻群演算法-粒子群演算法，結合基因演算法的全域搜尋能力、螞蟻群演算法的歷史記憶引導與粒子群演算法的動態速度調整，提升求解效率與穩定性，並改善解易陷入局部極小值之問題。最終成功反算出九個獨立正交材料常數，並代回理論模型與有限元素法模擬進行正向驗證，證實本研究所採最佳化方法在材料常數反算上之準確性與可行性。

This study proposes a non-destructive identification method for orthotropic material constants of rectangular plates with free boundaries, based on resonance frequencies and mode shapes. Experimentally, specimens with solid and grid-periodic structures were fabricated using fused deposition modeling 3D printing in three different printing directions. The specimens were excited using a loudspeaker, and their resonance frequencies were measured via laser Doppler vibrometer, while corresponding mode shapes were captured using electronic speckle pattern interferometry. Theoretically, the analytical framework was developed by integrating orthotropic material constitutive relations with classical thin plate assumptions. The governing equation for out-of-plane vibration was solved using a superposition method to satisfy specific free boundary conditions. The displacement functions were expressed in orthogonal form to analytically solve the eigenvalue problem, yielding the resonance frequencies and mode shapes, which were then cross-validated with finite element method (FEM) simulations. To inversely determine nine independent orthotropic material constants, a genetic algorithm (GA) was initially adopted. However, due to its tendency to become trapped in local minima and premature convergence in high-dimensional, multi-modal nonlinear problems, an adaptive simulated annealing mechanism was incorporated to enhance search capabilities. Despite improvements, global exploration and stability remained limited. To address this, a hybrid optimization strategy combining GA, ant colony optimization (ACO), and particle swarm optimization (PSO) was ultimately proposed. This hybrid approach integrates the global search ability of GA, the historical memory guidance of ACO, and the dynamic velocity adjustment of PSO, significantly improving the convergence efficiency and stability while avoiding local optima. The final identified material constants were successfully validated through forward analyses using the theoretical model and FEM simulations, confirming the accuracy and feasibility of the proposed optimization framework for inverse identification of orthotropic material properties.