整合體素法與有限單元法於薄壁加工之形變分析

Abstract

本研究首先復刻並驗證既有的體素模型與有限單元法（Finite Cell Method, FCM）的整合框架，過程中發現切削力與形變計算存在不匹配的現象。為解決此問題，本研究進一步提出改良之迭代耦合策略，以有效處理薄壁零件銑削加工中「切削力–形變耦合」的預測難題。 研究方法方面，首先利用STL模型建立八元樹結構的體素模型，透過三階段碰撞檢測（AABB、Plane–AABB 與 SAT）有效地完成工件的幾何離散化；接著以有限單元法於嵌入域內計算工件的剛性，並利用積分子網格細化邊界，確保模型計算的精度。同時，本研究採用局部剛性更新策略，僅針對受材料移除影響的積分網格進行重組，以降低計算成本。在切削力的計算方面，採用七參數通用模型描述刀具輪廓與螺旋刃，並配合切削範圍表格快速判定刀具與工件之間的接觸區域，透過切屑厚度的計算進一步推導切向、徑向及軸向分力。 此外，針對傳統「先力後變形」方法所產生的計算不一致與震盪現象，本研究引入Aitken–放鬆混合迭代演算法，每一插補點同步更新切削力與局部剛度矩陣，直至Dice係數與位移場誤差皆達收斂標準為止。 模擬驗證結果顯示，在相同負載條件下，本模型對最大節點位移的預測誤差約為 6.6%，與 SolidWorks® 有限元素分析的結果具有良好的一致性。此外，針對壁厚、切深及切寬等參數進行的模擬實驗，其觀察到的變形趨勢皆符合理論預期，進一步驗證了本模型的有效性與合理性。最後，透過與實際加工實驗之比較，亦確認了本研究所提出模型的可靠性，證明其具有實務應用價值，而非僅止於理論推演。 綜合上述，本研究建立的「體素FCM 耦合形變分析系統」可作為薄壁零件加工參數優化與靜態形變預測的基礎。未來研究將進一步拓展至多尺度網格，並將所提出的迭代耦合框架延伸至動態形變與顫振分析，探索切削加工的動態特性，以實現完整動態加工模擬的終極目標。

This study first replicates and verifies an existing coupled framework integrating the voxel-based modeling approach with the Finite Cell Method (FCM). During this process, discrepancies between cutting force and deformation calculations were identified. To address this issue, an improved iterative coupling strategy is proposed to effectively handle the "cutting force-deformation coupling" problem encountered in milling thin-walled parts. Regarding the methodology, an octree-based voxel model is initially generated from an STL file, and a three-stage collision detection (AABB, Plane–AABB, and SAT) is implemented to efficiently discretize the workpiece geometry. Subsequently, the FCM is applied within the embedded domain to compute the stiffness of the workpiece, while integral sub-cell refinement at boundaries ensures computational accuracy. Additionally, this study employs a partial stiffness matrix updating approach, reassembling the integral mesh only in regions affected by material removal, significantly reducing computational cost. To calculate the cutting forces, a generalized seven-parameter model describing tool profiles and helical cutting edges is adopted, coupled with a rapid determination method of the Cutter-Workpiece Engagement (CWE) based on a pre-constructed engagement table. The tangential, radial, and axial cutting forces are then computed through the undeformed chip thickness. Furthermore, addressing the inconsistencies and oscillations in traditional "force-before-deformation" methods, an Aitken relaxation-based iterative algorithm is introduced. At each interpolation point, cutting forces and local stiffness matrices are updated simultaneously until both the Dice coefficient and displacement field errors meet convergence criteria. Simulation results indicate that under identical loading conditions, the proposed model's maximum node displacement predictions exhibit a 6.6% error compared with SolidWorks® finite element analysis, demonstrating good agreement. Additional parametric studies on varying wall thicknesses, depths of cut, and widths of cut yielded deformation trends consistent with theoretical expectations, further validating the model's effectiveness and accuracy. Finally, through comparisons with actual machining experiments, the reliability of the developed model is confirmed, proving its practical applicability beyond theoretical validation. In summary, the established "Voxel-FCM coupled deformation analysis system" in this study can serve as a foundational tool for optimizing machining parameters and predicting static deformation in thin-walled components. Future research will extend this approach to multi-scale meshes and dynamic deformation and chatter analyses, exploring dynamic characteristics in milling processes, with the ultimate goal of achieving comprehensive dynamic machining simulations.