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研究生： 謝政廷
Hsieh, Cheng-Ting
論文名稱： GPU加速RKPM無網格法於大變形問題之演算法設計與實作
A GPU-Accelerated Meshfree RKPM Framework for Large-Deformation Simulation
指導教授： 林冠中
Lin, Kuan-Chung
學位類別： 碩士
Master
系所名稱： 工學院 - 土木工程學系
Department of Civil Engineering
論文出版年： 2026
畢業學年度： 114
語文別： 中文
論文頁數： 153
中文關鍵詞： 再生核粒子法 、GPU 平行運算 、大變形分析 、自然穩定化非合規節點積分 、半拉格朗日 、MEGA
外文關鍵詞： Reproducing Kernel Particle Method (RKPM), GPU Parallel Computing, Large Deformation Analysis, Naturally Stabilized Non-conforming Nodal Integration (NSNNI), Semi-Lagrangian, MEGA
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    • 土木與大地防災工程中之邊坡滑動、土石流衝擊與極端衝擊載荷等大變形問題，常使傳統有限元素法 (Finite Element Method, FEM) 因網格嚴重扭曲而陷入計算發散之困境。再生核粒子法 (Reproducing Kernel Particle Method, RKPM) 等無網格方法透過離散節點克服此一拓撲限制，是解決大變形與材料破壞問題之理想工具。然而，無網格法於每一時間步皆需執行鄰近節點搜尋與形函數動態建構，計算成本極為高昂，單次完整分析動輒需時數小時甚至數日，嚴重限制其於實際工程設計與大規模災害風險評估中之應用潛力。
    為突破此一計算效率瓶頸，本研究於 MEGA (Meshfree Explicit Galerkin Analysis)程式架構之上，開發一套圖形處理器 (Graphics Processing Unit, GPU) 加速之高效能無網格演算法。理論面採用半拉格朗日 (Semi-Lagrangian) 架構動態重建形函數，徹底避免變形梯度映射失效之問題，並引入自然穩定化非合規節點積分 (Naturally Stabilized Non-conforming Nodal Integration, NSNNI) 兼顧穩定性與變分一致性。實作面透過 NVIDIA CUDA 將計算密集模組移植至 GPU，配合原子操作、合併記憶體存取與多串流之 CPU–GPU 異質協同策略，最大化系統整體吞吐量。
    本研究以 Taylor bar 高速衝擊與堤防滑動（二維與三維）兩案例進行驗證。模擬結果與物理試驗及物質點法 (MPM) 高度吻合，且 GPU 與 CPU 版本之數值誤差於 0.1% 以內。效能方面，GPU 版本相較單核心 CPU 序列版本最高達成 17.5 倍加速，將原本需數小時之分析縮短至 20 分鐘以內。本研究所建立之 GPU 加速 RKPM 框架兼具數值穩定性與計算效率，其多串流機制更可於單張 GPU 同時執行數十至數百組獨立模擬，為大規模即時性之工程災害模擬與參數研究提供可行之技術路徑。

    In civil and geotechnical engineering, extreme large deformation phenomena such as slope failures, debris flow impacts, and high-velocity penetration problems often cause the traditional Finite Element Method (FEM) to suffer from computational divergence due to severe mesh distortion. Meshfree methods, in particular the Reproducing Kernel Particle Method (RKPM), overcome these topological constraints by discretizing the problem domain with a set of scattered nodes; however, the prohibitive computational cost—often requiring hours per simulation due to frequent neighbor searching and dynamic shape function construction—severely limits their applicability to large-scale engineering practice and real-time hazard assessment.
    To overcome this efficiency bottleneck, this study develops a GPU- accelerated meshfree framework on the basis of the MEGA (Meshfree Explicit Galerkin Analysis) code. A Semi-Lagrangian description is adopted to dynamically reconstruct shape functions at each time step, eliminating deformation gradient mapping failures, while the Naturally Stabilized Non-conforming Nodal Integration (NSNNI) is introduced to ensure numerical stability and variational consistency without user-tuned parameters. The compute-intensive modules are ported to NVIDIA CUDA with parallel strategies tailored to the Single Instruction Multiple Threads (SIMT) architecture, including atomicAdd-based force assembly, coalesced memory access, and a multi-stream pipeline for concurrent CPU–GPU heterogeneous computing.
    The proposed framework is validated through two extreme large deformation benchmarks: the Taylor bar high-velocity impact and the levee landslide problem in both two- and threedimensional geometries. The simulation results capture the mushrooming plastic deformation of the metallic bar and the large-scale fluid-like soil flow of the granular slope, with deviations from the CPU sequential reference solutions confined within 0.1%. In terms of computational performance, the GPU implementation achieves a peak speedup of 17.5× relative to the single-threaded CPU baseline, reducing simulation times that would take several hours to tens of minutes. This framework establishes a scalable foundation for real-time large-scale engineering disaster simulations and parametric reliability studies.

    中文摘要 I Abstract II 誌謝 IX 目錄 XI 表目錄 XVIII 圖目錄 XX 符號說明 XXII 第一章 緒論 1 1-1. 研究動機 1 1-2. 文獻回顧 3 1-2.1 再生核粒子法之發展與穩定化節點積分 3 1-2.2 大變形理論架構與半拉格朗日策略 4 1-2.3 GPU 平行運算於數值方法之發展 5 1-2.4 既有研究之缺口與本研究之定位 6 1-3. 本文結構 7 第二章 無網格數值方法 8 2-1. 再生核粒子近似之基本架構 8 2-1.1 離散化與場變數近似 8 2-1.2 形函數之建構 9 2-1.3 核函數與支撐域 10 2-1.4 形函數之一階與二階梯度 11 2-2. 隱式 RK 梯度與梯度展開 12 2-2.1 隱式 RK 梯度之公式 12 2-2.2 隱式梯度展開 13 2-3. Galerkin 弱形式之節點積分 14 2-3.1 模型問題與弱形式 14 2-3.2 直接節點積分 (DNI) 15 2-3.3 再生核梯度平滑方法 16 2-3.4 穩定化合規節點積分 (SCNI) 18 2-3.5 穩定化非合規節點積分 (SNNI) 19 2-4. 節點積分中之數值不穩定性分析 20 2-5. 自然穩定節點積分 (NSNI) 22 2-5.1 隱式梯度展開於應變之應用 22 2-5.2 NSNI 之雙線性形式 24 2-6. 大變形理論架構 26 2-6.1 更新拉格朗日架構 26 2-6.2 有限應變問題之運動學與基本方程式 26 2-6.3 更新拉格朗日與全拉格朗日之比較 27 2-7. 拉格朗日與半拉格朗日 RK 實作 28 2-7.1 拉格朗日再生核粒子近似 28 2-7.2 半拉格朗日再生核粒子近似 30 2-7.3 邊界奇異核處理本質邊界條件 31 第三章 平行運算理論 32 3-1. CPU 與 GPU 架構之比較 32 3-1.1 設計哲學差異 32 3-1.2 多核與眾核架構 33 3-2. CUDA 平行運算模型 34 3-2.1 階層式運算模型 35 3-2.2 GPU 記憶體階層 37 3-3. GPU 計算之效能特徵與瓶頸 38 3-3.1 延遲隱藏機制 38 3-3.2 序列化執行與分支發散 39 3-3.3 CPU 與 GPU 之資料傳輸瓶頸 40 3-4. 競爭條件與平行組裝策略 41 3-4.1 競爭條件之發生與原子操作 41 3-4.2 圖著色法 42 3-4.3 執行緒分配策略 42 3-5. 平行運算理論與本研究實作之對應 44 第四章 GPU 加速無網格 RKPM 程式撰寫說明 45 4-1. MEGA 程式架構與計算模組分解 45 4-2. 前處理與鄰近節點搜尋 48 4-2.1 鄰近搜尋與平滑域之數學定義 49 4-2.2 CPU 端循序鄰近搜尋 50 4-2.3 GPU 端多階段平行鄰近搜尋 50 4-2.4 前處理之雙版本實作比較 51 4-3. 再生核形狀函數 52 4-3.1 即時求值之 CPU 形狀函數計算 52 4-3.2 預先計算之 GPU 形狀函數計算 53 4-3.3 形狀函數計算之雙版本實作比較 54 4-4. 內力計算 55 4-4.1 客觀應力積分與內力組裝公式 55 4-4.2 單體迴圈之 CPU 內力實作 56 4-4.3 細粒度 kernel 之 GPU 內力實作 57 4-4.4 內力計算之雙版本實作比較 58 4-5. 外力計算 59 4-5.1 體積力與重力之離散公式 59 4-5.2 CPU 之內嵌式外力累加 60 4-5.3 GPU 之專用外力 kernel 60 4-6. 全域力向量組裝 61 4-6.1 散佈式組裝與集中質量公式 61 4-6.2 thread-private 暫存歸約策略 61 4-6.3 atomicAdd 原子累加策略 62 4-6.4 組裝策略之雙版本實作比較 62 4-7. 運動方程式與時間積分 63 4-7.1 中央差分法與預測校正公式 63 4-7.2 CPU 主程式之時間迴圈 64 4-7.3 GPU 之 Main kernel 序列 65 4-7.4 時間積分之雙版本實作比較 66 4-8. CPU 與 GPU 兩版本之系統性對應 67 第五章 數值模擬 68 5-1. 驗證案例之選擇與測試規劃 68 5-2. 測試環境 69 5-3. Taylor bar 衝擊 71 5-3.1 問題描述 71 5-3.2 幾何模型與離散化 71 5-3.3 材料參數 72 5-3.4 邊界條件與初始條件 74 5-3.5 再生核近似與時間積分設定 74 5-3.6 Lagrangian 模擬結果 76 5-3.7 Semi-Lagrangian 模擬結果 79 5-4. 邊坡滑動 82 5-4.1 問題描述 82 5-4.2 Drucker–Prager 塑性本構律 83 5-4.3 二維邊坡滑動：物理試驗對照 85 5-4.3.1 幾何模型與離散化 85 5-4.3.2 材料參數 86 5-4.3.3 邊界條件、初始條件與重力 86 5-4.3.4 再生核近似與時間積分設定 87 5-4.3.5 模擬結果與實驗對照 88 5-4.4 二維邊坡滑動：節點積分方法比較 90 5-4.4.1 幾何模型與離散化 90 5-4.4.2 材料參數 91 5-4.4.3 邊界條件、初始條件與重力 92 5-4.4.4 再生核近似與時間積分設定 92 5-4.4.5 模擬結果 93 5-4.4.6 計算效能 96 5-4.5 三維邊坡滑動 97 5-4.5.1 幾何模型與離散化 97 5-4.5.2 材料參數 97 5-4.5.3 邊界條件、初始條件與重力 98 5-4.5.4 再生核近似與時間積分設定 99 5-4.5.5 模擬結果 100 5-4.5.6 計算效能 102 5-5. 綜合精度與效能評估 103 5-5.1 計算精度驗證 103 5-5.2 加速比綜合比較 105 5-5.3 適用範圍與限制 106 第六章 建議與未來展望 108 6-1. 本研究之主要研究結論 108 6-1.1 GPU 加速效益顯著且維持數值精度 108 6-1.2 建立 CPU–GPU 協同計算框架 109 6-1.3 多流平行批次模擬之應用潛力 110 6-1.4 整體研究貢獻 110 6-2. 後續研究方向之具體建議 111 6-2.1 水土相互作用之引入 111 6-2.2 流固耦合模型與降雨入滲分析 112 6-2.3 滲流分析與不同飽和條件下之破壞機制 113 6-2.4 GPU 計算技術之延伸發展 114 6-2.4.1 混合精度計算與 Tensor Core 加速 114 6-2.4.2 多 GPU MPI 混合平行架構 114 6-2.5 災害模擬平台之長期發展藍圖 115 參考文獻 116

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