Multi-axis milling simulation plays a critical role in tool path verification and material removal analysis for complex part manufacturing. This paper presents a two-level voxel grid framework that enables efficient workpiece update and cutter-workpiece engagement determination under high-resolution conditions. The direct voxel tracing algorithm, designed for efficient localized updates on the uniform grid, is extended to two-level uniform/adaptive grids to balance efficiency and memory usage, but suffers degradation when applied naively due to hierarchical access overhead and redundant computation. To address this, a dynamic computation localization strategy is proposed, comprising three predictive mechanisms: an oriented bounding cylinder for direction-aware spatial filtering, a predictive engagement region strategy to reduce unnecessary computation, and a dual-stage Boolean reduction process to compress the voxel update domain. These mechanisms collectively confine computation to regions with actual cutter-workpiece interaction, systematically reducing computational cost. Comparative studies demonstrate the superior performance of the proposed method over uniform grid and tri-dexel representations under high-resolution conditions.
LasemiAXueDGuP.Recent development in CNC machining of freeform surfaces: a state-of-the-art review. Comput-Aided Des2010; 42(7): 641–654.
2.
GuoQLiuZYangZ, et al. Development, challenges and future trends on the fabrication of micro-textured surfaces using milling technology. J Manuf Process2024; 126: 285–331.
3.
GongHWangN.Analytical calculation of the envelope surface for generic milling tools directly from CL-data based on the moving frame method. Comput-Aided Des2009; 41(11): 848–855.
4.
JoyJFengHY.Frame-sliced voxel representation: an accurate and memory-efficient modeling method for workpiece geometry in machining simulation. Comput-Aided Des2017; 88: 1–13.
5.
JoyJFengHY.Efficient milling part geometry computation via three-step update of frame-sliced voxel representation workpiece model. Int J Adv Manuf Technol2017; 92(5-8): 2365–2378.
6.
AltasEGokkayaHKaratasM, et al. Analysis of surface roughness and flank wear using the Taguchi method in milling of NiTi shape memory alloy with uncoated tools. Coatings2020; 10(12): 1259.
7.
AltasEAltin KaratasMGokkayaH, et al. Surface integrity of NiTi shape memory alloy in milling with cryogenic heat treated cutting tools under different cutting conditions. J Mater Eng Perform2021; 30: 9426–9439.
8.
AltasEErkanOOzkanD, et al. Optimization of cutting conditions, parameters, and cryogenic heat treatment for surface roughness in milling of NiTi shape memory alloy. J Mater Eng Perform2022; 31: 7315–7327.
9.
SiHWangL.Error compensation in the five-axis flank milling of thin-walled workpieces. Proc IMechE, Part B: J Engineering Manufacture2019; 233(4): 1224–1234.
10.
YangYWuDLiuQ.Chatter stability prediction of milling considering nonlinearities. Proc IMechE, Part B: J Engineering Manufacture2021; 235(5): 862–876.
11.
XiongBWeiYGuD, et al. Chatter stability analysis of variable speed milling with helix angled cutters. Proc IMechE, Part B: J Engineering Manufacture2021; 235(5): 850–861.
12.
LiZLWangXZZhuLM.Arc–surface intersection method to calculate cutter–workpiece engagements for generic cutter in five-axis milling. Comput-Aided Des2016; 73: 1–10.
13.
MaengSRBaekNShinSY, et al. A Z-map update method for linearly moving tools. Comput-Aided Des2003; 35(11): 995–1009.
14.
TunçLTOzkirimliOMBudakE.Machining strategy development and parameter selection in 5-axis milling based on process simulations. Int J Adv Manuf Technol2016; 85: 1483–1500.
15.
ZhangXYuTWangW.Modeling, simulation, and optimization of five-axis milling processes. Int J Adv Manuf Technol2014; 74: 1611–1624.
16.
SchnösFHartmannDObstB, et al. GPU accelerated voxel-based machining simulation. Int J Adv Manuf Technol2021; 115: 275–289.
17.
NieZFengHY.Integrated and efficient cutter-workpiece engagement determination in three-axis milling via voxel modeling. Int J Adv Manuf Technol2023; 128: 391–403.
18.
KukrejaADhandaMPandeSS.Voxel-based adaptive toolpath planning using graphics processing unit for freeform surface machining. J Manuf Sci Eng2022; 144(1): 011013.
LiYTangKZengL.A voxel model-based process-planning method for five-axis machining of complicated parts. J Comput Inf Sci Eng2020; 20(4): 041012.
21.
MiersJCTuckerTKurfessT, et al. Voxel-based modeling of transient material removal in machining. Int J Adv Manuf Technol2021; 116: 1575–1589.
22.
NieZFengHY.Efficient voxel-based workpiece update and cutter-workpiece engagement determination in multi-axis milling. J Manuf Sci Eng2024; 146(6): 061003.
23.
MaHLiuWZhouX, et al. High efficiency calculation of cutter-workpiece engagement in five-axis milling using distance fields and envelope theory. J Manuf Process2021; 68: 1430–1447.
24.
NieZZhaoY.High-accuracy and efficient simulation of numerical control machining using tri-level grid and envelope theory. Machines2025; 13(1): 69.
25.
HossainMMNathCTuckerTM, et al. A graphics processor unit-accelerated freeform surface offsetting method for high-resolution subtractive three-dimensional printing (machining). J Manuf Sci Eng2018; 140(4): 041012.
26.
YousefianOBalabokhinATarbuttonJ.Point-by-point prediction of cutting force in 3-axis CNC milling machines through voxel framework in digital manufacturing. J Intell Manuf2020; 31: 215–226.
27.
WouSJShinYCEl-MounayriH.Ball end milling mechanistic model based on a voxel-based geometric representation and a ray casting technique. J Manuf Process2013; 15(3): 338–347.
28.
WittMSchumannMKlimantP.Real-time machine simulation using cutting force calculation based on a voxel material removal model. Int J Adv Manuf Technol2019; 105: 2321–2328.
29.
NishidaIOkumuraRSatoR, et al. Cutting force simulation in minute time resolution for ball end milling under various tool posture. J Manuf Sci Eng2018; 140(2): 021009.
30.
KanekoKShimizuJShiraseK.A voxel-based end milling simulation method to analyze the elastic deformation of a workpiece. J Manuf Sci Eng2023; 145(1): 011005.
