本研究應用離散元素法（discrete element method，DEM）於結構複合實驗(hybrid simulation)中分析數值子結構。傳統的有限元素法雖具有廣泛的適用性以及可靠性，進行大型結構的非線性動力歷時分析時，除了求解運動方程式的計算量大，為求分析模擬的準確性，常需要在一步的計算步長中進行多次的迭代，過程過於繁瑣。此外，有限元素法不容易進行結構大變形乃至於倒塌的模擬。有鑑於此，本研究旨在探索可適用於複合實驗的數值分析方法，應用離散元素法於複合實驗中，以擴展複合實驗應用的廣度與深度。
離散元素法大多應用在土壤及岩石力學中，本研究初步探討如何應用離散元素法於結構分析上；此外，不同時間步長的選擇可能導致不穩定的離散元素分析結果，本研究使用能進行一次迭代(有預測項和修正項)的α-OS積分法，並探討調整α值造成結構線性動力分析結果的差異性。最後，以雙線性模型配合α-OS積分法，成功且穩定地完成了離散元素非線性複合實驗的模擬，確保在進行真實複合實驗時的穩定性和安全性。
由於離散元素法將結構自由度拆分成獨立的粒子，並且各粒子之間的計算是獨立的，為了更有效使用此離散元素法的特性，本研究導入平行運算，以任務並行的方式，在不同處理器核心處理不同任務時，彼此之間也能進行數據的交換，以滿足結構各層樓獨立計算的過程中存在數據的相依性，成功且穩定地結合平行運算與離散元素法於複合實驗的模擬。
最後，為了驗證離散元素法與平行運算於結構複合實驗的可行性，本研究分別選擇兩層樓與九層樓結構進行複合實驗技術的驗證，其中第一層樓做為實驗子結構，使用鋼棒與鉸支承提供所需之側向恢復力，當離散元素法於每一積分步得到的第一層樓與地表的相對位移後，將此位移作為油壓致動器的位移命令施加於試體上，並量測試體的恢復力回傳給數值子結構。實驗結果顯示，本研究可成功且穩定地完成應用離散元素法的複合實驗；此外，結合平行運算與離散元素法之複合實驗結果與僅使用離散元素法之結果差異甚小。未來可應用平行運算與離散元素法進行較複雜的非線性複合實驗。

This study utilizes the Discrete Element Method (DEM) for structural analysis. While the traditional Finite Element Method (FEM) is widely applicable and highly reliable, analyzing nonlinear structural behavior using FEM requires multiple iterations within a single time step, resulting in longer computational times. Therefore, the aim of this research is to explore a new approach in hybrid simulation by employing the Discrete Element Method. This approach aims to address certain challenges that cannot be resolved using FEM in hybrid simulation, and can be resolved through the application of the DEM.
Due to the prevalent application of the Discrete Element Method (DEM) in soil and rock mechanics, this study initially explores the application of DEM in structural analysis. As the stability of DEM results can be affected by the choice of time step, the α-OS integration method is selected to perform a single iteration (with a predictor and corrector) to mitigate this issue. By adjusting the α value to vary the numerical damping ratio, the results of numerical simulations for linear behavior are compared with those obtained from state space analysis. Once the differences are deemed negligible, it is confirmed that this method is suitable for analyzing linear structural behavior.
For the nonlinear portion, the α-OS integration method is combined with a bilinear model. The Symbiotic Organisms Search (SOS) algorithm is employed to optimize the nonlinear model parameters of the coupons. Subsequently, the analysis is conducted to predict the nonlinear behavior of the structure, ensuring stability and safety during nonlinear experiments. This approach enables the control of the entire experimental process.
To enhance the efficient application of the Discrete Element Method (DEM), parallel computing is introduced in this study. Since DEM involves the subdivision of structural degrees of freedom into individual particles that are independently calculated, this concept aligns well with parallel computing. However, there is data dependency among the calculations of different floors during the independent computation process. Therefore, this research adopts a task parallelism approach where different processors handle different tasks and exchange data with each other. This integration of parallel computing and the DEM enables efficient and effective analysis.
This study utilizes hybrid simulation to validate the application of the Discrete Element Method (DEM) and parallel computing in analyzing the seismic response of structures. Coupons and hinge supports are employed to provide lateral stiffness. The structural models used in this study consist of a two-story structure model and a nine-story structure model. The first floor is considered as the experimental substructure, while the structure above the first floor is treated as the numerical substructure. The relative displacement between the first-floor structure and the ground, obtained from each integration step, is used as input for the hydraulic actuators. The measured restoring forces from the actuators are then fed back to the numerical substructure. The experimental results demonstrate the feasibility of using the DEM in hybrid simulation. Moreover, the incorporation of parallel computing has minimal impact on the analysis results of the DEM. Therefore, the combination of parallel computing and the DEM remains viable.

參考文獻
1. Dong, X., Z. Tang, and X. Du, State of the art and development trends in numerical simulation for real-time hybrid simulation. Earthquake Engineering and Resilience, 2022. 1(3): p. 245-267.
2. Saouma, V., D.-H. Kang, and G. Haussmann, A computational finite-element program for hybrid simulation. Earthquake Engineering & Structural Dynamics, 2012. 41(3): p. 375-389.
3. Carrion, J.S., Billie, Model-based Strategies for Real-time Hybrid Testing. 2007, University of Illinois at Urbana-Champaign, The Newmark Structural Engineering Laboratory. p. 21-25.
4. Chen, C. and J.M. Ricles, Analysis of implicit HHT-α integration algorithm for real-time hybrid simulation. Earthquake Engineering & Structural Dynamics, 2012. 41(5): p. 1021-1041.
5. Combescure, D. and P. Pegon, α-Operator splitting time integration technique for pseudodynamic testing error propagation analysis. Soil Dynamics and Earthquake Engineering, 1997. 16(7): p. 427-443.
6. Chen, C., et al., Real-time hybrid testing using the unconditionally stable explicit CR integration algorithm. Earthquake Engineering & Structural Dynamics, 2009. 38(1): p. 23-44.
7. Nakashima, M., H. Kato, and E. Takaoka, Development of real-time pseudo dynamic testing. Earthquake Engineering & Structural Dynamics, 1992. 21(1): p. 79-92.
8. Bonnet, P.A., et al., Real-time hybrid experiments with Newmark integration, MCSmd outer-loop control and multi-tasking strategies. Earthquake Engineering & Structural Dynamics, 2007. 36(1): p. 119-141.
9. Cundall, P.A. and O.D.L. Strack, A discrete numerical model for granular assemblies. Géotechnique, 1979. 29(1): p. 47-65.
10. Munjiza, A., T. Bangash, and N.W.M. John, The combined finite–discrete element method for structural failure and collapse. Engineering Fracture Mechanics, 2004. 71(4): p. 469-483.
11. Suchorzewski, J., J. Tejchman, and M. Nitka, Discrete element method simulations of fracture in concrete under uniaxial compression based on its real internal structure. International Journal of Damage Mechanics, 2018. 27(4): p. 578-607.
12. Di, S. and S. Liu, Analysis of ice load on conical structure with discrete element method. Engineering Computations, 2015. 32: p. 1121-1134.
13. Duan, Y., et al., Real-time hybrid simulation based on vector form intrinsic finite element and field programmable gate array. Structural Control and Health Monitoring, 2019. 26(1): p. e2277.
14. Lu, L.-Q., J.-T. Wang, and F. Zhu, Improvement of Real-Time Hybrid Simulation Using Parallel Finite-Element Program. Journal of Earthquake Engineering, 2020. 24(10): p. 1547-1565.
15. Furumura, T., K. Koketsu, and K.L. Wen, Parallel PSM/FDM Hybrid Simulation of Ground Motions from the 1999 Chi-Chi, Taiwan, Earthquake. pure and applied geophysics, 2002. 159(9): p. 2133-2146.