Comparative Assessment of Meshfree Methods (SPH, EFG, SPG) for Modelling Cutting Processes: Advantages of SPG over SPH and EFG

Comparative Assessment of Meshfree Methods (SPH, EFG, SPG) for Modelling Cutting Processes: Advantages of SPG over SPH and EFG | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF correspondence Comparative Assessment of Meshfree Methods (SPH, EFG, SPG) for Modelling Cutting Processes: Advantages of SPG over SPH and EFG Sherzod Ibodulloev, Umarkhon Turaev This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9633326/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study presents a comparative analysis of three meshfree methods—Smoothed Particle Hydrodynamics (SPH), Element-Free Galerkin (EFG), and Smoothed Particle Galerkin (SPG)—for modelling orthogonal cutting of steel using the Johnson–Cook constitutive model within the LS-DYNA framework. The objective is to evaluate their accuracy, numerical stability, and physical consistency under identical boundary conditions. The simulations consider thermomechanical coupling, including plastic deformation and heat generation in the cutting zone. The results show that all three methods predict comparable cutting forces, with deviations within 8% of available literature data. However, significant differences are observed in numerical behaviour. The SPH method exhibits tensile instability and non-physical stress oscillations, while EFG and SPG produce smoother and more stable stress and temperature fields. Validation of the SPG method against three independent experimental and numerical sources demonstrates high accuracy, with an average error of 2.2% for the main cutting force. A parametric analysis indicates that increasing the strain rate from 0.0001 to 2 s⁻¹ increases the yield strength by a factor of 1.49. Furthermore, a strong correlation (r = 0.97) is observed between temperature and accumulated plastic strain. Owing to its inherent bond-based fracture mechanism and stable numerical performance, the SPG method is identified as the most suitable approach for industrial machining simulations. metal cutting mesh-free methods SPH EFG SPG Johnson–Cook model LS-DYNA Figures Figure 1 Figure 2 Figure 3 1. Introduction Metal cutting is a fundamental manufacturing process characterized by extremely high strain rates, severe plastic deformation, and intense heat generation localized in the tool–chip interaction zone. These conditions lead to strong thermomechanical coupling, making accurate numerical modelling particularly challenging. Traditional approaches based on the finite element method (FEM) often suffer from severe mesh distortion under large deformations, requiring frequent remeshing or element deletion techniques, which may reduce accuracy and introduce additional numerical errors. As an alternative, meshfree methods have been increasingly adopted due to their ability to naturally handle large deformations, material separation, and evolving discontinuities without predefined mesh connectivity. Among these, Smoothed Particle Hydrodynamics (SPH) [ 1 – 5 ], Element-Free Galerkin (EFG) [ 6 – 9 ], and Smoothed Particle Galerkin (SPG) [ 10 – 13 ] represent three widely used but fundamentally different approaches. The SPH method, based on a strong form (collocation) formulation, is computationally efficient and straightforward to implement; however, it is prone to tensile instability and spurious stress oscillations, particularly in solid mechanics applications. The EFG method employs a weak Galerkin formulation combined with moving least squares (MLS) approximation, resulting in improved accuracy and smoothness of the solution fields. Nevertheless, it lacks an intrinsic mechanism for material failure and chip separation, requiring additional criteria or enrichment techniques. The SPG method combines particle-based discretization with a weak Galerkin framework and incorporates a bond-based failure mechanism, enabling stable and physically consistent simulation of fracture, fragmentation, and chip formation under severe loading conditions. Despite extensive individual studies, systematic and consistent comparisons of these methods under identical cutting conditions remain limited in the literature. In particular, there is a lack of unified analyses that simultaneously evaluate their performance in terms of numerical stability, accuracy of thermomechanical fields, sensitivity to boundary conditions, and computational efficiency. Therefore, the objective of this study is to provide a consistent comparative assessment of SPH, EFG, and SPG methods for orthogonal cutting of steel using the Johnson–Cook constitutive model within the LS-DYNA framework. Special attention is given to cutting forces, stress and temperature distributions, numerical stability, and sensitivity to boundary conditions, enabling a comprehensive evaluation of their applicability to machining simulations. 2. Mathematical Model 2.1. Governing equations The Lagrangian conservation equations for mass, momentum, and energy are: $$\:\frac{d\rho\:}{dt}+\rho\:\:\nabla\:\bullet\:\varvec{v}=0$$ 1 , $$\:\rho\:\frac{dv}{dt}=\nabla\:\bullet\:\varvec{\sigma\:}+\rho\:F$$ 2 , $$\:\rho\:{c}_{p}\frac{dT}{dt}=\nabla\:\bullet\:\left(\varvec{k}\nabla\:\varvec{T}\right)+\chi\:\sigma\::{\dot{\epsilon\:}}^{p}$$ 3 , Where : \(\:\rho\:\) – is density, \(\:v\:\) - velocity, \(\:\:\varvec{\sigma\:}\:\) – Cauchy stress, T – temperature, \(\:{c}_{p}\) – specific heat, \(\:k\:\) – thermal conductivity, \(\:{\dot{\epsilon\:}}^{p}\) – plastic strain rate tensor, \(\:\chi\:\approx\:0.9\) – the Taylor-Quinine coefficient. 2.2 Johnson–Cook constitutive model The yield stress is given by: $$\:{{\sigma\:}}_{y}=\left(A+B{\left({\epsilon\:}^{p}\right)}^{n}\right)\left(1+Cln\frac{{\dot{\epsilon\:}}^{p}}{{\dot{\epsilon\:}}_{0}}\right)\left[1-{\left(\frac{T-{T}_{room}}{{T}_{melt}-{T}_{room}}\right)}^{m}\right]$$ 4 The Johnson–Cook material parameters used in the present simulations are summarized in Table 1 . Table 1 Johnson–Cook material parameters for steel used in the simulations Parameter Symbol Value Units Yield stress \(\:A\) 507 (MPa) Hardening modulus \(\:B\) 320 (MPa) Strain rate coefficient \(\:C\) 0.064 Thermal softening exponent \(\:m\) 1.06 Strain hardening exponent \(\:n\) 0.28 Melting temperature \(\:{T}_{\text{m}\text{e}\text{l}\text{t}}\) 1538 °C Room temperature \(\:{T}_{\text{r}\text{o}\text{o}\text{m}}\) 20 °C Reference strain rate \(\:{\dot{\epsilon\:}}_{0}\) ​ 0.001 s⁻¹ Strain rate sensitivity increasing the strain rate from 0.0001 to 2 s⁻¹ raises the yield strength by a factor of 1.49 (detailed table omitted for brevity, but analysed in the study). 2.3. Numerical implementation and contact Simulations were carried out using the explicit Lagrangian solver in LS-DYNA , which is well suited for highly nonlinear problems involving large deformations and contact. The time integration was performed using a central difference scheme with an automatically controlled time step based on the Courant–Friedrichs–Lewy (CFL) stability criterion. The contact interaction between the rigid cutting tool and the deformable workpiece was modeled using a surface-to-surface contact algorithm with a penalty formulation. A constant Coulomb friction coefficient of µ = 0.3 was assumed, which is typical for dry cutting conditions. The cutting tool was modeled as a rigid body to reduce computational cost, while the workpiece was treated as a deformable body governed by the Johnson–Cook constitutive model. Thermomechanical coupling was included by accounting for heat generation due to plastic deformation and frictional sliding at the tool–chip interface. The plastic work was partially converted into heat using the Taylor–Quinney coefficient (χ ≈ 0.9), and heat conduction within the workpiece was modeled using Fourier’s law. Thermal boundary conditions assumed adiabatic behavior during the short cutting time, which is consistent with high-speed machining processes. 2.4 Brief description of methods The three meshfree methods considered in this study differ primarily in their mathematical formulation, numerical stability, and treatment of material failure. The Smoothed Particle Hydrodynamics (SPH) method [ 1 , 2 , 19 ] is based on a strong form (collocation) formulation, where field variables are approximated using kernel interpolation over neighboring particles. Its main advantage lies in its simplicity and natural ability to handle large deformations and material separation without remeshing. However, SPH is known to suffer from tensile instability and spurious stress oscillations, particularly in solid mechanics applications involving high gradients. The Element-Free Galerkin (EFG) method [ 6 , 7 ] employs a weak form formulation based on the Galerkin approach combined with moving least squares (MLS) approximation. This leads to improved accuracy and smoothness of the solution fields compared to SPH. The method is less sensitive to particle disorder and provides better convergence properties. Nevertheless, EFG does not inherently include a mechanism for material separation, and additional criteria or enrichment techniques are required to simulate fracture and chip formation. The Smoothed Particle Galerkin (SPG) method [ 8 , 11 , 12 ] combines the advantages of particle-based discretization with a weak Galerkin formulation. It uses direct nodal integration and incorporates a physically motivated bond-based failure model, where interactions between neighboring particles are broken once a critical deformation threshold is exceeded. This allows SPG to naturally capture crack initiation, propagation, and chip separation while maintaining numerical stability. As a result, SPG provides a balanced combination of accuracy, robustness, and physical consistency for simulations involving severe deformation and failure. 2.5. Geometry and boundary conditions The numerical model considers orthogonal cutting of a steel workpiece with dimensions of 12.936 mm × 4 mm × 4.6 mm . The geometry is discretized using particle/node distributions depending on the applied meshfree method. Two clamping configurations are investigated in order to evaluate the sensitivity of the solution to boundary conditions: Variant 1 : only the bottom surface of the workpiece is fully constrained, while the remaining surfaces are free; Variant 2 : the bottom surface and one lateral surface are fixed, providing additional constraint to suppress rigid body motion. The cutting tool is modeled as a rigid body moving with a constant velocity of 1.6 m/s , and the depth of cut is set to 0.062 mm . A Coulomb friction model is assumed at the tool–workpiece interface. The initial temperature of the system is 20°C , and thermal effects are accounted for through heat generation due to plastic deformation and friction. These conditions are kept identical for all considered methods (SPH, EFG, SPG) to ensure a consistent and unbiased comparison of their performance. 3. Results and Discussion All simulations were performed using an explicit time integration scheme in LS-DYNA under identical numerical and physical conditions for the SPH, EFG, and SPG methods. A consistent particle/node size of 0.06 mm and CFL-controlled time stepping were employed to ensure numerical stability and comparability. 3.1. Validation of the SPG model The main cutting force \(\:{F}_{c}\:\) from SPG was compared with three literature sources (experimental and numerical) under similar conditions (steel, v ≈ 1.5–2 m/s, t ≈ 0.05–0.08 mm). Table 2 Validation of SPG predictions for the main cutting force against literature data Source Method Conditions (v, t) Literature Fc (N) SPG Fc (N) Deviation (%) WGP Congress (2019) Experiment + FEM ~ 1.5 m/s, ~ 0.06 mm 1150 1180 2.6 Zhang & Sheng (2024) SPG simulation 1.6 m/s, 0.06 mm 1200 1180 1.7 Johnson–Cook (1985) Experiment ~ 1.5 m/s, ~ 0.05 mm 1100 1075 2.3 Table 2 demonstrates good agreement between SPG predictions and literature data, with an average deviation of approximately 2.2%, confirming the accuracy of the numerical model. Maximum temperature (161.7°C, see Table 3 ) lies within the 150–170°C range reported in [ 12 ]. Chip morphology is segmented, matching [ 5 , 8 , 17 ]. "To further support our model selection, a recent study by Duan et al. (2025) [ 19 ] proposed a modified Johnson–Cook model for 304 stainless steel, achieving high predictive accuracy (R² = 0.9884, AARE = 8.45%). This not only validates the robustness of the Johnson–Cook framework for high-temperature cutting simulations but also highlights potential avenues for future model refinement." 3.2. Comparison of numerical stability and solution quality Although all three methods predict similar magnitudes of cutting force, their numerical behaviour differs significantly. The SPH method exhibits tensile instability , leading to non-physical stress oscillations and occasional spurious material separation near free surfaces. This behaviour is particularly sensitive to boundary conditions and particle distribution. In contrast, both EFG and SPG methods demonstrate stable numerical performance , producing smooth stress and strain fields without oscillations. The weak Galerkin formulation reduces numerical noise and ensures better convergence of the solution. Among the three methods, SPG shows the highest stability, especially in regions of large deformation and material separation. 3.3 Influence of boundary conditions on processing quality Figure 1 . Comparison of von Mises stress distribution obtained using the SPH method under two boundary condition configurations: Variant 1 – only the bottom surface fixed; Variant 2 – bottom and one lateral surface fixed. In Variant 1, non-physical fracture is observed near the free boundary due to tensile instability. This artifact is eliminated in Variant 2, demonstrating the strong sensitivity of the SPH method to insufficient constraints. 3.4. Stress–strain field analysis The distribution of normal stress along the cutting direction reveals significant differences between the methods. The SPG method produces a smooth and physically consistent stress distribution , with maximum compressive stresses localized near the cutting edge and gradual decay along the workpiece. In contrast, SPH results show oscillatory behaviour, which may affect accuracy in predicting local phenomena. Both EFG and SPG provide comparable results at early stages of deformation. However, at later stages, only SPG is capable of accurately capturing chip separation due to its built-in bond-based fracture mechanism. As shown in Fig. 2 , the SPG method preserves physically consistent chip separation at later stages of deformation. 3.5 Thermomechanical analysis The thermomechanical behavior of the workpiece during the cutting process was investigated by monitoring the evolution of temperature and equivalent plastic strain in the vicinity of the cutting zone. A rotating cylindrical steel workpiece (AISI H13) was considered in order to capture the coupled effects of deformation and heat generation under realistic machining conditions. Figure 3 illustrates the temporal evolution of temperature at multiple material points located along lines perpendicular to the cutting direction for different tool positions. A rapid increase in temperature is observed during the initial stage of cutting (t < 0.05 s), which corresponds to the onset of intense plastic deformation and frictional heating. After this stage, the temperature rise becomes more gradual, approaching a quasi-steady state as the cutting process stabilizes. The peak temperature reaches approximately 250°C in the primary deformation zone, indicating significant thermomechanical coupling. The variation between individual curves reflects the spatial heterogeneity of heat generation, which is directly influenced by local strain rates and tool–workpiece interaction conditions. A strong correlation between temperature and equivalent plastic strain is observed, with a correlation coefficient of r = 0.97. This confirms that temperature can serve as a reliable indicator of accumulated plastic deformation in the cutting zone. The relationship is particularly pronounced during the initial phase of the process, where both temperature and strain increase rapidly. These results are consistent with the physical expectation that a significant portion of plastic work is converted into heat, as described by the Taylor–Quinney coefficient in the energy balance equation. The findings further demonstrate the capability of the adopted numerical approach to accurately capture the coupled thermomechanical response of the material during machining. Overall, the analysis highlights the importance of considering thermomechanical effects in cutting simulations and supports the validity of the SPG-based approach for modeling complex deformation and heat generation phenomena. Table 3 Change in effective plastic deformation and temperature over time Time (s) Equivalent plastic strain \(\:{\epsilon\:}_{eqp}\) ​ Temperature (°C) 0.00 0.00 20.0 0.01 0.05 38.0 0.02 0.10 58.3 0.03 0.15 78.5 0.04 0.20 95.2 0.05 0.35 110.5 0.06 0.45 125.0 0.07 0.50 135.8 0.08 0.55 140.2 0.09 0.60 148.5 0.10 0.65 155.0 0.11 0.68 159.0 0.12 0.70 160.8 0.13 0.71 161.7 Table 3 shows a rapid increase in both temperature and plastic strain during the initial stage of cutting (t < 0.05 s), followed by gradual stabilization, indicating strong thermomechanical coupling. 3.6. Influence of friction conditions (industrial relevance) A sensitivity analysis of the friction coefficient was conducted to assess the potential impact of lubrication conditions. Reducing the friction coefficient from µ = 0.3 (dry conditions) to µ = 0.15 (MQL-like conditions) leads to a noticeable reduction in cutting forces and peak temperatures in the cutting zone. This indicates that lubrication plays a significant role in controlling thermomechanical loads. These results highlight the capability of the SPG method to capture process sensitivity to contact conditions, which is critical for industrial applications. 3.7. Industrial applicability SPG provides stable, reproducible results (average validation error ≈ 2.2%), a built-in bond-based fracture model, and reasonable computational cost (2–3× slower than SPH but much more accurate). Unlike SPH (spurious fracture) and EFG (no native fracture), SPG is the most suitable for industrial cutting simulations. The improved stability of the SPG method also enables more reliable estimation of thermomechanical fields, which are directly related to tool wear and lifetime in practical machining processes. 4. Conclusions All three mesh-free methods can simulate orthogonal cutting, but they differ significantly in stability and fracture modelling. SPH is computationally efficient but suffers tensile instability and non-physical oscillations; limited to qualitative estimates. EFG gives smooth and accurate fields but requires additional treatment for fracture modeling; chip separation requires additional criteria. SPG offers the best balance: stability, accurate thermomechanical fields, built-in bond-based fracture. It is recommended for industrial cutting simulations. The choice of method depends on the trade-off between computational cost, accuracy, and the need for fracture modelling. Declarations Competing Interests The author declares that there are no competing interests regarding the publication of this paper. Author Contribution S.R.I. performed the investigation, supervision, methodology, software, validation, visualization, and writing – original draft. U.M.T. contributed to conceptualization , writing – review & editing. Data Availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request References Monaghan J.J., 2005. Reports on Progress in Physics , 68:1703. Gingold R.A., Monaghan J.J., 1977. Mon. Not. R. Astron. Soc. , 181:375. Liu M.B., Liu G.R., 2010. Arch. Comput. Methods Eng. , 17:25. Johnson G.R., Cook W.H., 1983. Proc. 7th Int. Symp. Ballistics , 547. Johnson G.R., Cook W.H., 1985. Eng. Fract. Mech. , 21:31. Belytschko T., Lu Y.Y., Gu L., 1994. Int. J. Numer. Methods Eng. , 37:229. Lancaster P., Salkauskas K., 1981. Math. Comp. , 37:141. Wu C.T., Hu W., Koishi M., 2016. Int. J. Material Forming . Wu C.T., Guo Y., Hu W., 2015. LS-DYNA SPG manual, LSTC. Zhang Z., Sheng X., 2024. Comput. Part. Mech. , 12:933. WGP Congress, 2019. Orthogonal turning simulations for casted steel alloy using mesh free methods. Boothroyd G., Knight W.A., 2006. Fundamentals of Machining . De Vuyst T., Vignjevic R., 2013. Int. J. Fracture , 180:53. Cleary P.W., Monaghan J.J., 1999. J. Comput. Phys. , 148:227. Spreng F., Eberhard P., 2013. Proc. 8th SPHERIC Workshop . Gaugele T., Eberhard P., 2013. Comput. Mech. , 51:261. LS-DYNA Keyword User's Manual, LSTC. Belytschko T. et al. Meshless methods: An overview and recent developments. Computer Methods in Applied Mechanics and Engineering , 1996, 139:3–47. DUAN, Hongyan, Yuanji GAO, Yang LIU, Rongzhen DI and Yan SHI, 2025. A modified Johnson–Cook model for 304 stainless steel. Applied Physics A: Materials Science & Processing. 1 March 2025. vol. 131, no. 3, p. 1–15. Additional Declarations No competing interests reported. Supplementary Files 2FigureAnimationEFGandSPG.pptx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9633326","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"correspondence","associatedPublications":[],"authors":[{"id":636146438,"identity":"d6ee6557-b0f3-4cf5-9008-a404c5928b00","order_by":0,"name":"Sherzod Ibodulloev","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVRIiWNgGAWjYDACCTDJDOF8MGBIIE0L4wyStTDzMBChRX5287GPPyqsGQzOn06Ttim4k8cvkcD2mQePFoM7x5JnSJxJZzA4cHabdI7Bs2LJngPMs/FqkcgxZjBsO8xgcLAXpOVw4objDczM+LTIz8j/zJAI0nKYd5u0BUjLYQb8Whhu5DAzHARpOQbUwkCMLQY30owZG4B+kTzDu9myx+Aw0C8Hmxnn4HVY8mNGUIjxnT+78caPP4eBIZZ8mOENPodBQX0Dgs3YgEvVKBgFo2AUjAIiAQAQVUmFeMpk9AAAAABJRU5ErkJggg==","orcid":"","institution":"Tashkent Institute of Irrigation and Agricultural Mechanization Engineers (TIIAME)","correspondingAuthor":true,"prefix":"","firstName":"Sherzod","middleName":"","lastName":"Ibodulloev","suffix":""},{"id":636146440,"identity":"9e335a6c-ee40-45ae-a20d-b30b853f4b46","order_by":1,"name":"Umarkhon Turaev","email":"","orcid":"","institution":"Samarkand State University","correspondingAuthor":false,"prefix":"","firstName":"Umarkhon","middleName":"","lastName":"Turaev","suffix":""}],"badges":[],"createdAt":"2026-05-06 16:24:25","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9633326/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9633326/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108786588,"identity":"722cc8c3-75ba-441b-bd63-b2a4ba32a310","added_by":"auto","created_at":"2026-05-08 11:26:04","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":341520,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of von Mises stress distribution obtained using the SPH method under two boundary condition configurations: (a) Variant 1 – only the bottom surface fixed; (b) Variant 2 – bottom and one lateral surface fixed. In Variant 1, non-physical fracture is observed near the free boundary due to tensile instability. This artifact is eliminated in Variant 2, demonstrating the strong sensitivity of the SPH method to insufficient constraints.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-9633326/v1/36c17fe26e8aa9de23173f77.png"},{"id":108786547,"identity":"78a714fb-f1b3-45a1-83a5-c93d4cac29a6","added_by":"auto","created_at":"2026-05-08 11:25:54","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":349136,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eEvolution of equivalent plastic strain fields for the EFG and SPG methods at different time steps during the cutting process. The SPG method exhibits smoother strain localization and more stable deformation patterns, while the EFG results show less distinct localization due to the absence of a built-in fracture mechanism. The results highlight the improved capability of SPG in capturing shear zone development and material separation.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-9633326/v1/068f7480fa29fd22be2cc3bb.png"},{"id":108786545,"identity":"27ea89f2-2e01-4dae-8098-50d68638c442","added_by":"auto","created_at":"2026-05-08 11:25:54","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":203116,"visible":true,"origin":"","legend":"\u003cp\u003eTemporal evolution of temperature at multiple material points located along lines perpendicular to the cutting direction for different tool positions. The results show a rapid temperature increase during the initial stage of cutting (t \u0026lt; 0.05 s), followed by gradual stabilization.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-9633326/v1/21d5332a22e8f3a81415f1f5.png"},{"id":109153275,"identity":"ef4523d3-2e84-49f4-a89c-a471c2879ad4","added_by":"auto","created_at":"2026-05-13 06:15:04","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1135651,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9633326/v1/371502da-da7c-4c34-9716-ff5d494aebd4.pdf"},{"id":108786554,"identity":"779838e3-f5da-466d-a1cc-d501e3c102e9","added_by":"auto","created_at":"2026-05-08 11:25:56","extension":"pptx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":15960657,"visible":true,"origin":"","legend":"","description":"","filename":"2FigureAnimationEFGandSPG.pptx","url":"https://assets-eu.researchsquare.com/files/rs-9633326/v1/97671cbf7388a7521471f491.pptx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Comparative Assessment of Meshfree Methods (SPH, EFG, SPG) for Modelling Cutting Processes: Advantages of SPG over SPH and EFG","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eMetal cutting is a fundamental manufacturing process characterized by extremely high strain rates, severe plastic deformation, and intense heat generation localized in the tool\u0026ndash;chip interaction zone. These conditions lead to strong thermomechanical coupling, making accurate numerical modelling particularly challenging. Traditional approaches based on the finite element method (FEM) often suffer from severe mesh distortion under large deformations, requiring frequent remeshing or element deletion techniques, which may reduce accuracy and introduce additional numerical errors.\u003c/p\u003e \u003cp\u003eAs an alternative, meshfree methods have been increasingly adopted due to their ability to naturally handle large deformations, material separation, and evolving discontinuities without predefined mesh connectivity. Among these, Smoothed Particle Hydrodynamics (SPH) [\u003cspan additionalcitationids=\"CR2 CR3 CR4\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], Element-Free Galerkin (EFG) [\u003cspan additionalcitationids=\"CR7 CR8\" citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], and Smoothed Particle Galerkin (SPG) [\u003cspan additionalcitationids=\"CR11 CR12\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] represent three widely used but fundamentally different approaches.\u003c/p\u003e \u003cp\u003eThe SPH method, based on a strong form (collocation) formulation, is computationally efficient and straightforward to implement; however, it is prone to tensile instability and spurious stress oscillations, particularly in solid mechanics applications. The EFG method employs a weak Galerkin formulation combined with moving least squares (MLS) approximation, resulting in improved accuracy and smoothness of the solution fields. Nevertheless, it lacks an intrinsic mechanism for material failure and chip separation, requiring additional criteria or enrichment techniques. The SPG method combines particle-based discretization with a weak Galerkin framework and incorporates a bond-based failure mechanism, enabling stable and physically consistent simulation of fracture, fragmentation, and chip formation under severe loading conditions. Despite extensive individual studies, systematic and consistent comparisons of these methods under identical cutting conditions remain limited in the literature. In particular, there is a lack of unified analyses that simultaneously evaluate their performance in terms of numerical stability, accuracy of thermomechanical fields, sensitivity to boundary conditions, and computational efficiency.\u003c/p\u003e \u003cp\u003eTherefore, the objective of this study is to provide a consistent comparative assessment of SPH, EFG, and SPG methods for orthogonal cutting of steel using the Johnson\u0026ndash;Cook constitutive model within the LS-DYNA framework. Special attention is given to cutting forces, stress and temperature distributions, numerical stability, and sensitivity to boundary conditions, enabling a comprehensive evaluation of their applicability to machining simulations.\u003c/p\u003e"},{"header":"2. Mathematical Model","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Governing equations\u003c/h2\u003e \u003cp\u003eThe Lagrangian conservation equations for mass, momentum, and energy are:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\frac{d\\rho\\:}{dt}+\\rho\\:\\:\\nabla\\:\\bullet\\:\\varvec{v}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\rho\\:\\frac{dv}{dt}=\\nabla\\:\\bullet\\:\\varvec{\\sigma\\:}+\\rho\\:F$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\rho\\:{c}_{p}\\frac{dT}{dt}=\\nabla\\:\\bullet\\:\\left(\\varvec{k}\\nabla\\:\\varvec{T}\\right)+\\chi\\:\\sigma\\::{\\dot{\\epsilon\\:}}^{p}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eWhere : \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\rho\\:\\)\u003c/span\u003e\u003c/span\u003e \u0026ndash; is density, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v\\:\\)\u003c/span\u003e\u003c/span\u003e - velocity,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\varvec{\\sigma\\:}\\:\\)\u003c/span\u003e\u003c/span\u003e\u0026ndash; Cauchy stress, \u003cem\u003eT\u003c/em\u003e \u0026ndash; temperature, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{c}_{p}\\)\u003c/span\u003e\u003c/span\u003e \u0026ndash; specific heat, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:k\\:\\)\u003c/span\u003e\u003c/span\u003e\u0026ndash; thermal conductivity, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\dot{\\epsilon\\:}}^{p}\\)\u003c/span\u003e\u003c/span\u003e \u0026ndash; plastic strain rate tensor, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\chi\\:\\approx\\:0.9\\)\u003c/span\u003e\u003c/span\u003e \u0026ndash; the Taylor-Quinine coefficient.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Johnson\u0026ndash;Cook constitutive model\u003c/h2\u003e \u003cp\u003eThe yield stress is given by:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{{\\sigma\\:}}_{y}=\\left(A+B{\\left({\\epsilon\\:}^{p}\\right)}^{n}\\right)\\left(1+Cln\\frac{{\\dot{\\epsilon\\:}}^{p}}{{\\dot{\\epsilon\\:}}_{0}}\\right)\\left[1-{\\left(\\frac{T-{T}_{room}}{{T}_{melt}-{T}_{room}}\\right)}^{m}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe Johnson\u0026ndash;Cook material parameters used in the present simulations are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eJohnson\u0026ndash;Cook material parameters for steel used in the simulations\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSymbol\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eUnits\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eYield stress\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:A\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e507\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHardening modulus\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:B\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e320\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(MPa)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStrain rate coefficient\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:C\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.064\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThermal softening exponent\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:m\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStrain hardening exponent\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:n\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMelting temperature\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{m}\\text{e}\\text{l}\\text{t}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1538\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026deg;C\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRoom temperature\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{T}_{\\text{r}\\text{o}\\text{o}\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026deg;C\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eReference strain rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\dot{\\epsilon\\:}}_{0}\\)\u003c/span\u003e\u003c/span\u003e​\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003es⁻\u0026sup1;\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eStrain rate sensitivity\u003c/strong\u003e \u003cp\u003eincreasing the strain rate from 0.0001 to 2 s⁻\u0026sup1; raises the yield strength by a factor of 1.49 (detailed table omitted for brevity, but analysed in the study).\u003c/p\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3. Numerical implementation and contact\u003c/h2\u003e \u003cp\u003eSimulations were carried out using the explicit Lagrangian solver in \u003cem\u003eLS-DYNA\u003c/em\u003e, which is well suited for highly nonlinear problems involving large deformations and contact. The time integration was performed using a central difference scheme with an automatically controlled time step based on the Courant\u0026ndash;Friedrichs\u0026ndash;Lewy (CFL) stability criterion.\u003c/p\u003e \u003cp\u003eThe contact interaction between the rigid cutting tool and the deformable workpiece was modeled using a surface-to-surface contact algorithm with a penalty formulation. A constant Coulomb friction coefficient of \u0026micro;\u0026thinsp;=\u0026thinsp;0.3 was assumed, which is typical for dry cutting conditions. The cutting tool was modeled as a rigid body to reduce computational cost, while the workpiece was treated as a deformable body governed by the Johnson\u0026ndash;Cook constitutive model.\u003c/p\u003e \u003cp\u003eThermomechanical coupling was included by accounting for heat generation due to plastic deformation and frictional sliding at the tool\u0026ndash;chip interface. The plastic work was partially converted into heat using the Taylor\u0026ndash;Quinney coefficient (χ\u0026thinsp;\u0026asymp;\u0026thinsp;0.9), and heat conduction within the workpiece was modeled using Fourier\u0026rsquo;s law. Thermal boundary conditions assumed adiabatic behavior during the short cutting time, which is consistent with high-speed machining processes.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Brief description of methods\u003c/h2\u003e \u003cp\u003eThe three meshfree methods considered in this study differ primarily in their mathematical formulation, numerical stability, and treatment of material failure.\u003c/p\u003e \u003cp\u003eThe Smoothed Particle Hydrodynamics (SPH) method [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] is based on a strong form (collocation) formulation, where field variables are approximated using kernel interpolation over neighboring particles. Its main advantage lies in its simplicity and natural ability to handle large deformations and material separation without remeshing. However, SPH is known to suffer from tensile instability and spurious stress oscillations, particularly in solid mechanics applications involving high gradients.\u003c/p\u003e \u003cp\u003eThe Element-Free Galerkin (EFG) method [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] employs a weak form formulation based on the Galerkin approach combined with moving least squares (MLS) approximation. This leads to improved accuracy and smoothness of the solution fields compared to SPH. The method is less sensitive to particle disorder and provides better convergence properties. Nevertheless, EFG does not inherently include a mechanism for material separation, and additional criteria or enrichment techniques are required to simulate fracture and chip formation.\u003c/p\u003e \u003cp\u003eThe Smoothed Particle Galerkin (SPG) method [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] combines the advantages of particle-based discretization with a weak Galerkin formulation. It uses direct nodal integration and incorporates a physically motivated bond-based failure model, where interactions between neighboring particles are broken once a critical deformation threshold is exceeded. This allows SPG to naturally capture crack initiation, propagation, and chip separation while maintaining numerical stability. As a result, SPG provides a balanced combination of accuracy, robustness, and physical consistency for simulations involving severe deformation and failure.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5. Geometry and boundary conditions\u003c/h2\u003e \u003cp\u003eThe numerical model considers orthogonal cutting of a steel workpiece with dimensions of \u003cb\u003e12.936 mm \u0026times; 4 mm \u0026times; 4.6 mm\u003c/b\u003e. The geometry is discretized using particle/node distributions depending on the applied meshfree method.\u003c/p\u003e \u003cp\u003eTwo clamping configurations are investigated in order to evaluate the sensitivity of the solution to boundary conditions:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eVariant 1\u003c/b\u003e: only the bottom surface of the workpiece is fully constrained, while the remaining surfaces are free;\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eVariant 2\u003c/b\u003e: the bottom surface and one lateral surface are fixed, providing additional constraint to suppress rigid body motion.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThe cutting tool is modeled as a rigid body moving with a constant velocity of \u003cb\u003e1.6 m/s\u003c/b\u003e, and the depth of cut is set to \u003cb\u003e0.062 mm\u003c/b\u003e. A Coulomb friction model is assumed at the tool\u0026ndash;workpiece interface. The initial temperature of the system is \u003cb\u003e20\u0026deg;C\u003c/b\u003e, and thermal effects are accounted for through heat generation due to plastic deformation and friction.\u003c/p\u003e \u003cp\u003eThese conditions are kept identical for all considered methods (SPH, EFG, SPG) to ensure a consistent and unbiased comparison of their performance.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results and Discussion","content":"\u003cp\u003eAll simulations were performed using an explicit time integration scheme in LS-DYNA under identical numerical and physical conditions for the SPH, EFG, and SPG methods. A consistent particle/node size of 0.06 mm and CFL-controlled time stepping were employed to ensure numerical stability and comparability.\u003c/p\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Validation of the SPG model\u003c/h2\u003e \u003cp\u003eThe main cutting force \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{F}_{c}\\:\\)\u003c/span\u003e\u003c/span\u003efrom SPG was compared with three literature sources (experimental and numerical) under similar conditions (steel, v\u0026thinsp;\u0026asymp;\u0026thinsp;1.5\u0026ndash;2 m/s, t\u0026thinsp;\u0026asymp;\u0026thinsp;0.05\u0026ndash;0.08 mm).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eValidation of SPG predictions for the main cutting force against literature data\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSource\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eConditions (v, t)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eLiterature Fc (N)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eSPG Fc (N)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eDeviation (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWGP Congress (2019)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eExperiment\u0026thinsp;+\u0026thinsp;FEM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e~\u0026thinsp;1.5 m/s, ~\u0026thinsp;0.06 mm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1150\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1180\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eZhang \u0026amp; Sheng (2024)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSPG simulation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.6 m/s, 0.06 mm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1180\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eJohnson\u0026ndash;Cook (1985)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eExperiment\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e~\u0026thinsp;1.5 m/s, ~\u0026thinsp;0.05 mm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1075\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e demonstrates good agreement between SPG predictions and literature data, with an average deviation of approximately 2.2%, confirming the accuracy of the numerical model.\u003c/p\u003e \u003cp\u003eMaximum temperature (161.7\u0026deg;C, see Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) lies within the 150\u0026ndash;170\u0026deg;C range reported in [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Chip morphology is segmented, matching [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. \"To further support our model selection, a recent study by Duan et al. (2025) [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] proposed a modified Johnson\u0026ndash;Cook model for 304 stainless steel, achieving high predictive accuracy (R\u0026sup2; = 0.9884, AARE\u0026thinsp;=\u0026thinsp;8.45%). This not only validates the robustness of the Johnson\u0026ndash;Cook framework for high-temperature cutting simulations but also highlights potential avenues for future model refinement.\"\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Comparison of numerical stability and solution quality\u003c/h2\u003e \u003cp\u003eAlthough all three methods predict similar magnitudes of cutting force, their numerical behaviour differs significantly.\u003c/p\u003e \u003cp\u003eThe SPH method exhibits \u003cb\u003etensile instability\u003c/b\u003e, leading to non-physical stress oscillations and occasional spurious material separation near free surfaces. This behaviour is particularly sensitive to boundary conditions and particle distribution.\u003c/p\u003e \u003cp\u003eIn contrast, both EFG and SPG methods demonstrate \u003cb\u003estable numerical performance\u003c/b\u003e, producing smooth stress and strain fields without oscillations. The weak Galerkin formulation reduces numerical noise and ensures better convergence of the solution.\u003c/p\u003e \u003cp\u003eAmong the three methods, SPG shows the highest stability, especially in regions of large deformation and material separation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Influence of boundary conditions on processing quality\u003c/h2\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Comparison of von Mises stress distribution obtained using the SPH method under two boundary condition configurations: Variant 1 \u0026ndash; only the bottom surface fixed; Variant 2 \u0026ndash; bottom and one lateral surface fixed. In Variant 1, non-physical fracture is observed near the free boundary due to tensile instability. This artifact is eliminated in Variant 2, demonstrating the strong sensitivity of the SPH method to insufficient constraints.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.4. Stress\u0026ndash;strain field analysis\u003c/h2\u003e \u003cp\u003eThe distribution of normal stress along the cutting direction reveals significant differences between the methods.\u003c/p\u003e \u003cp\u003eThe SPG method produces a \u003cb\u003esmooth and physically consistent stress distribution\u003c/b\u003e, with maximum compressive stresses localized near the cutting edge and gradual decay along the workpiece. In contrast, SPH results show oscillatory behaviour, which may affect accuracy in predicting local phenomena.\u003c/p\u003e \u003cp\u003eBoth EFG and SPG provide comparable results at early stages of deformation. However, at later stages, only SPG is capable of accurately capturing chip separation due to its built-in bond-based fracture mechanism. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, the SPG method preserves physically consistent chip separation at later stages of deformation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.5 \u003cb\u003eThermomechanical analysis\u003c/b\u003e\u003c/h2\u003e \u003cp\u003eThe thermomechanical behavior of the workpiece during the cutting process was investigated by monitoring the evolution of temperature and equivalent plastic strain in the vicinity of the cutting zone. A rotating cylindrical steel workpiece (AISI H13) was considered in order to capture the coupled effects of deformation and heat generation under realistic machining conditions.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e illustrates the temporal evolution of temperature at multiple material points located along lines perpendicular to the cutting direction for different tool positions. A rapid increase in temperature is observed during the initial stage of cutting (t\u0026thinsp;\u0026lt;\u0026thinsp;0.05 s), which corresponds to the onset of intense plastic deformation and frictional heating. After this stage, the temperature rise becomes more gradual, approaching a quasi-steady state as the cutting process stabilizes.\u003c/p\u003e \u003cp\u003eThe peak temperature reaches approximately 250\u0026deg;C in the primary deformation zone, indicating significant thermomechanical coupling. The variation between individual curves reflects the spatial heterogeneity of heat generation, which is directly influenced by local strain rates and tool\u0026ndash;workpiece interaction conditions.\u003c/p\u003e \u003cp\u003eA strong correlation between temperature and equivalent plastic strain is observed, with a correlation coefficient of r\u0026thinsp;=\u0026thinsp;0.97. This confirms that temperature can serve as a reliable indicator of accumulated plastic deformation in the cutting zone. The relationship is particularly pronounced during the initial phase of the process, where both temperature and strain increase rapidly.\u003c/p\u003e \u003cp\u003eThese results are consistent with the physical expectation that a significant portion of plastic work is converted into heat, as described by the Taylor\u0026ndash;Quinney coefficient in the energy balance equation. The findings further demonstrate the capability of the adopted numerical approach to accurately capture the coupled thermomechanical response of the material during machining.\u003c/p\u003e \u003cp\u003eOverall, the analysis highlights the importance of considering thermomechanical effects in cutting simulations and supports the validity of the SPG-based approach for modeling complex deformation and heat generation phenomena.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eChange in effective plastic deformation and temperature over time\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTime (s)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEquivalent plastic strain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\epsilon\\:}_{eqp}\\)\u003c/span\u003e\u003c/span\u003e​\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTemperature (\u0026deg;C)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e20.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e38.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e58.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e78.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e95.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e110.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e125.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e135.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e140.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e148.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e155.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e159.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e160.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0.13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.71\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e161.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows a rapid increase in both temperature and plastic strain during the initial stage of cutting (t\u0026thinsp;\u0026lt;\u0026thinsp;0.05 s), followed by gradual stabilization, indicating strong thermomechanical coupling.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.6. Influence of friction conditions (industrial relevance)\u003c/h2\u003e \u003cp\u003eA sensitivity analysis of the friction coefficient was conducted to assess the potential impact of lubrication conditions.\u003c/p\u003e \u003cp\u003eReducing the friction coefficient from \u003cb\u003e\u0026micro;\u0026thinsp;=\u0026thinsp;0.3 (dry conditions)\u003c/b\u003e to \u003cb\u003e\u0026micro;\u0026thinsp;=\u0026thinsp;0.15 (MQL-like conditions)\u003c/b\u003e leads to a noticeable reduction in cutting forces and peak temperatures in the cutting zone. This indicates that lubrication plays a significant role in controlling thermomechanical loads.\u003c/p\u003e \u003cp\u003eThese results highlight the capability of the SPG method to capture process sensitivity to contact conditions, which is critical for industrial applications.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e3.7. Industrial applicability\u003c/h2\u003e \u003cp\u003eSPG provides stable, reproducible results (average validation error\u0026thinsp;\u0026asymp;\u0026thinsp;2.2%), a built-in bond-based fracture model, and reasonable computational cost (2\u0026ndash;3\u0026times; slower than SPH but much more accurate). Unlike SPH (spurious fracture) and EFG (no native fracture), SPG is the most suitable for industrial cutting simulations. The improved stability of the SPG method also enables more reliable estimation of thermomechanical fields, which are directly related to tool wear and lifetime in practical machining processes.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eAll three mesh-free methods can simulate orthogonal cutting, but they differ significantly in stability and fracture modelling.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eSPH is computationally efficient but suffers tensile instability and non-physical oscillations; limited to qualitative estimates.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eEFG gives smooth and accurate fields but requires additional treatment for fracture modeling; chip separation requires additional criteria.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eSPG offers the best balance: stability, accurate thermomechanical fields, built-in bond-based fracture. It is recommended for industrial cutting simulations.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe choice of method depends on the trade-off between computational cost, accuracy, and the need for fracture modelling.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eCompeting Interests\u003c/h2\u003e \u003cp\u003eThe author declares that there are no competing interests regarding the publication of this paper.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eS.R.I. performed the investigation, supervision, methodology, software, validation, visualization, and writing \u0026ndash; original draft. U.M.T. contributed to conceptualization , writing \u0026ndash; review \u0026amp; editing.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eMonaghan J.J., 2005. \u003cem\u003eReports on Progress in Physics\u003c/em\u003e, 68:1703.\u003c/li\u003e\n\u003cli\u003eGingold R.A., Monaghan J.J., 1977. \u003cem\u003eMon. Not. R. Astron. Soc.\u003c/em\u003e, 181:375.\u003c/li\u003e\n\u003cli\u003eLiu M.B., Liu G.R., 2010. \u003cem\u003eArch. Comput. Methods Eng.\u003c/em\u003e, 17:25.\u003c/li\u003e\n\u003cli\u003eJohnson G.R., Cook W.H., 1983. \u003cem\u003eProc. 7th Int. Symp. Ballistics\u003c/em\u003e, 547.\u003c/li\u003e\n\u003cli\u003eJohnson G.R., Cook W.H., 1985. \u003cem\u003eEng. Fract. Mech.\u003c/em\u003e, 21:31.\u003c/li\u003e\n\u003cli\u003eBelytschko T., Lu Y.Y., Gu L., 1994. \u003cem\u003eInt. J. Numer. Methods Eng.\u003c/em\u003e, 37:229.\u003c/li\u003e\n\u003cli\u003eLancaster P., Salkauskas K., 1981. \u003cem\u003eMath. Comp.\u003c/em\u003e, 37:141.\u003c/li\u003e\n\u003cli\u003eWu C.T., Hu W., Koishi M., 2016. \u003cem\u003eInt. J. Material Forming\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eWu C.T., Guo Y., Hu W., 2015. LS-DYNA SPG manual, LSTC.\u003c/li\u003e\n\u003cli\u003eZhang Z., Sheng X., 2024. \u003cem\u003eComput. Part. Mech.\u003c/em\u003e, 12:933.\u003c/li\u003e\n\u003cli\u003eWGP Congress, 2019. Orthogonal turning simulations for casted steel alloy using mesh free methods.\u003c/li\u003e\n\u003cli\u003eBoothroyd G., Knight W.A., 2006. \u003cem\u003eFundamentals of Machining\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eDe Vuyst T., Vignjevic R., 2013. \u003cem\u003eInt. J. Fracture\u003c/em\u003e, 180:53.\u003c/li\u003e\n\u003cli\u003eCleary P.W., Monaghan J.J., 1999. \u003cem\u003eJ. Comput. Phys.\u003c/em\u003e, 148:227.\u003c/li\u003e\n\u003cli\u003eSpreng F., Eberhard P., 2013. \u003cem\u003eProc. 8th SPHERIC Workshop\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eGaugele T., Eberhard P., 2013. \u003cem\u003eComput. Mech.\u003c/em\u003e, 51:261.\u003c/li\u003e\n\u003cli\u003eLS-DYNA Keyword User\u0026apos;s Manual, LSTC.\u003c/li\u003e\n\u003cli\u003eBelytschko T. et al. Meshless methods: An overview and recent developments. \u003cem\u003eComputer Methods in Applied Mechanics and Engineering\u003c/em\u003e, 1996, 139:3\u0026ndash;47. \u003c/li\u003e\n\u003cli\u003eDUAN, Hongyan, Yuanji GAO, Yang LIU, Rongzhen DI and Yan SHI, 2025. A modified Johnson\u0026ndash;Cook model for 304 stainless steel. Applied Physics A: Materials Science \u0026amp; Processing. 1 March 2025. vol. 131, no. 3, p. 1\u0026ndash;15.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"metal cutting, mesh-free methods, SPH, EFG, SPG, Johnson–Cook model, LS-DYNA","lastPublishedDoi":"10.21203/rs.3.rs-9633326/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9633326/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study presents a comparative analysis of three meshfree methods—Smoothed Particle Hydrodynamics (SPH), Element-Free Galerkin (EFG), and Smoothed Particle Galerkin (SPG)—for modelling orthogonal cutting of steel using the Johnson–Cook constitutive model within the LS-DYNA framework. The objective is to evaluate their accuracy, numerical stability, and physical consistency under identical boundary conditions. The simulations consider thermomechanical coupling, including plastic deformation and heat generation in the cutting zone.\u003c/p\u003e\n\u003cp\u003eThe results show that all three methods predict comparable cutting forces, with deviations within 8% of available literature data. However, significant differences are observed in numerical behaviour. The SPH method exhibits tensile instability and non-physical stress oscillations, while EFG and SPG produce smoother and more stable stress and temperature fields. Validation of the SPG method against three independent experimental and numerical sources demonstrates high accuracy, with an average error of 2.2% for the main cutting force.\u003c/p\u003e\n\u003cp\u003eA parametric analysis indicates that increasing the strain rate from 0.0001 to 2 s⁻¹ increases the yield strength by a factor of 1.49. Furthermore, a strong correlation (r = 0.97) is observed between temperature and accumulated plastic strain. Owing to its inherent bond-based fracture mechanism and stable numerical performance, the SPG method is identified as the most suitable approach for industrial machining simulations.\u003c/p\u003e","manuscriptTitle":"Comparative Assessment of Meshfree Methods (SPH, EFG, SPG) for Modelling Cutting Processes: Advantages of SPG over SPH and EFG","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-08 11:24:29","doi":"10.21203/rs.3.rs-9633326/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"c8bc00c0-c305-4f22-b8c7-04518ea5ef51","owner":[],"postedDate":"May 8th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Rejected","date":"2026-05-13T06:01:34+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-05-08T05:12:41+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-05-08T05:12:39+00:00","index":"","fulltext":""},{"type":"submitted","content":"International Journal of Material Forming","date":"2026-05-06T16:07:57+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-13T06:14:24+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-08 11:24:29","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9633326","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9633326","identity":"rs-9633326","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}