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A Comparative Study on Two Mixed Least Squares Meshless Models with Improved SPH, MPS and CPM Methods to Solve Elasticity Problems

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Abstract

This paper investigates the accuracy of several meshless methods to solve elasticity problems. The methods include the well-known smoothed particle hydrodynamics (SPH) (called model 1 in this study), and Voronoi-based SPH (model 2), moving particle semi-implicit (MPS) (model 3), and Voronoi-based MPS (model 4) methods and three different proposed least squares models based on Taylor series expansion (TSE). The accuracy of both the SPH and MPS methods is improved by employing the Voronoi diagram as an alternative approach to estimate the computational node volume parameter. One of the least squares methods (model 5) uses TSE truncated to the second-order to solve the standard quadratic differential equations of elasticity problems, considering displacements as unknown variables. The two last methods employ the first-order (model 6) and second-order (model 7) TSE to approximate the function in the mixed formulation, where the governing equations can be written as a system of the first-order differential equations with unknown variables of stresses and displacements. The mixed formulation improves the prediction accuracy of unknown parameters, especially stress, by eliminating the second derivative calculations. The results indicate that the least squares methods, particularly model 5, can achieve higher accuracy and computational efficiency than SPH and MPS methods, especially in stress calculations. Noteworthy, the second-order mixed model exhibits considerable superiority over the first-order model while requiring approximately the same computational effort.

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References

  • Afshar MH, Amani J, Naisipour M (2012) A node enrichment adaptive refinement in Discrete Least Squares Meshless method for solution of elasticity problems. Eng Anal Boundary Elem 36(3):385–393

    Article  MathSciNet  Google Scholar 

  • Ahmadi I (2021) Vibration analysis of 2D-functionally graded nanobeams using the nonlocal theory and meshless method. Eng Anal Boundary Elem 124:142–154

    Article  MathSciNet  Google Scholar 

  • Alshenawy R, Sahmani S, Safaei B, Elmoghazy Y, Al-Alwan A, Al NM (2023) Surface stress effect on nonlinear dynamical performance of nanobeam-type piezoelectric energy harvesters via meshless collocation technique. Eng Anal Boundary Elem 152:104–119

    Article  MathSciNet  Google Scholar 

  • Amani J, Afshar MH, Naisipour M (2012) Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions. Eng Anal Boundary Elem 36(5):894–902

    Article  MathSciNet  Google Scholar 

  • Ataie-Ashtiani B, Shobeyri G (2008) Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics. Int J Numer Methods Fluids 56(2):209–232

    Article  MathSciNet  Google Scholar 

  • Atluri SN, Liu HT, Han ZD (2006) Meshless local Petrov–Galerkin (MLPG) mixed collocation method for elasticity problems. CMC-Tech Science Press 4(3):141

    MathSciNet  Google Scholar 

  • Choudhary K, DeCost B, Chen C, Jain A, Tavazza F, Cohn R, Park CW, Choudhary A, Agrawal A, Billinge SJ, Holm E (2022) Recent advances and applications of deep learning methods in materials science. npj Comput Mater 8(1):59

    Article  Google Scholar 

  • Duan G, Koshizuka S, Yamaji A, Chen B, Li X, Tamai T (2018) An accurate and stable multiphase moving particle semi-implicit method based on a corrective matrix for all particle interaction models. Int J Numer Methods Eng 115(10):1287–1314

    Article  MathSciNet  Google Scholar 

  • Duan G, Yamaji A, Koshizuka S, Chen B (2019) The truncation and stabilization error in multiphase moving particle semi-implicit method based on corrective matrix: which is dominant. Comput Fluids 190:254–273

    Article  MathSciNet  Google Scholar 

  • Gomes CP, Selman B, Gregoire JM (2019) Artificial intelligence for materials discovery. MRS Bull 44(7):538–544

    Article  Google Scholar 

  • Heydari Z, Shobeyri G, Ghoreishi Najafabadi SH (2020) Accuracy analysis of different higher-order Laplacian models of Incompressible SPH method. Eng Comput 37(1):181–202

    Article  Google Scholar 

  • Hirschler M, Huber M, Säckel W, Kunz P, Nieken U (2014) An application of the Cahn–Hilliard approach to smoothed particle hydrodynamics. Math Probl Eng 2014:694894

    Article  MathSciNet  Google Scholar 

  • Hu XY, Nikolaus NA (2007) An incompressible multi-phase SPH method. J Comput Phys 227(1):264–278

    Article  Google Scholar 

  • Huang C, Lei JM, Liu MB, Peng XY (2016) An improved KGF-SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows. Int J Numer Methods Fluids 81(6):377–396

    Article  MathSciNet  Google Scholar 

  • Jia Z, Wang W, Zhang J, Li H (2022) Contact high-temperature strain automatic calibration and precision compensation research. J Artif Intell Technol 2(2):69–76

    Google Scholar 

  • Khayyer A, Gotoh H (2010) A higher order Laplacian model for enhancement and stabilization of pressure calculation by the MPS method. Appl Ocean Res 32(1):124–131

    Article  Google Scholar 

  • Khayyer A, Gotoh H (2011) Enhancement of stability and accuracy of the moving particle semi-implicit method. J Comput Phys 230(8):3093–3118

    Article  MathSciNet  Google Scholar 

  • Khayyer A, Gotoh H (2013) Enhancement of performance and stability of MPS mesh-free particle method for multiphase flows characterized by high density ratios. J Comput Phys 242:211–233

    Article  MathSciNet  Google Scholar 

  • Kiani K (2014) A nonlocal meshless solution for flexural vibrations of double-walled carbon nanotubes. Appl Math Comput 234:557–578

    MathSciNet  Google Scholar 

  • Kiani K, Nikkhoo A, Mehri B (2009) Parametric analyses of multispan viscoelastic shear deformable beams under excitation of a moving mass. J Vib Acoust 131(5):051009

    Article  Google Scholar 

  • Kiani K, Shodja HM (2011) Prediction of the penetrated rust into the microcracks of concrete caused by reinforcement corrosion. Appl Math Model 35(5):2529–2543

    Article  MathSciNet  Google Scholar 

  • Kiani K, Shodja HM (2012) Response of reinforced concrete structures to macrocell corrosion of reinforcements. Part II: After propagation of microcracks via a numerical approach. Nucl Eng Des 242:7–18

    Article  Google Scholar 

  • Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123(3):421–434

    Article  Google Scholar 

  • Koh CG, Gao M, Luo C (2012) A new particle method for simulation of incompressible free surface flow problems. Int J Numer Methods Eng 89(12):1582–1604

    Article  MathSciNet  Google Scholar 

  • Li SC, Cheng YM (2005) Enriched meshless manifold method for two- dimensional crack modeling. Theoret Appl Fract Mech 44(3):234–248

    Article  Google Scholar 

  • Liang D, Jian W, Shao S, Chen R, Yang K (2017) Incompressible SPH simulation of solitary wave interaction with movable seawalls. J Fluids Struct 69:72–88

    Article  Google Scholar 

  • Luo M, Reeve DE, Shao S, Karunarathna H, Lin P, Cai H (2019) Consistent particle method simulation of solitary wave impinging on and overtopping a seawall. Eng Anal Boundary Elem 103:160–171

    Article  MathSciNet  Google Scholar 

  • Murotani K, Koshizuka S, Tamai T, Shibata K, Mitsume N, Yoshimura S, Fujisawa T (2014) Development of hierarchical domain decomposition explicit MPS method and application to large-scale tsunami analysis with floating objects. J Adv Simul Sci Eng 1(1):16–35

    Google Scholar 

  • Nikravesh KS, Afshar MH, Faraji S (2016) RPIM and RPIM-MLS based MDLSM method for the solution of elasticity problems. Sci Iran 23(6):2458–2468

    Google Scholar 

  • Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved SPH method: towards higher order convergence. J Comput Phys 225(2):1472–1492

    Article  MathSciNet  Google Scholar 

  • Ran Q, Tong J, Shao S, Fu X, Xu Y (2015) Incompressible SPH scour model for movable bed dam break flows. Adv Water Resour 82:39–50

    Article  Google Scholar 

  • Renxiang L, Weiwen Z, Decheng W (2023) Numerical simulation of solid-liquid mixed transport flow in ultra-long pipelines by MPS-DEM method. Chin J Comput Mech

  • Sha W, Guo Y, Yuan Q, Tang S, Zhang X, Lu S, Guo X, Cao YC, Cheng S (2020) Artificial intelligence to power the future of materials science and engineering. Adv Intell Syst 2(4):1900143

    Article  Google Scholar 

  • Shamsirband S, Mehri KN (2021) Micro-mechanical damage diagnosis methodologies based on machine learning and deep learning models. J Zhejiang Univ SCIENCE A 22(8):585–608

    Article  Google Scholar 

  • Shao S, Lo EY (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26(7):787–800

    Article  Google Scholar 

  • Shimizu Y, Khayyer A, Gotoh H (2022) An enhanced incompressible SPH method for simulation of fluid flow interactions with saturated/unsaturated porous media of variable porosity. Ocean Syst Eng 12(1):63–86

    Google Scholar 

  • Schwaiger HF (2008) An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions. Int J Numer Methods Eng 75(6):647–671

    Article  MathSciNet  Google Scholar 

  • Shobeyri G, Ardakani RR (2017) Improving accuracy of SPH method using Voronoi diagram. Iran J Sci Technol Trans Civ Eng 41:345–350

    Article  Google Scholar 

  • Shobeyri G, Madadi H (2018) An improvement in MPS method using Voronoi diagram and a new kernel function. J Braz Soc Mech Sci Eng 40(4):1–10

    Article  Google Scholar 

  • Shobeyri G (2019) Improving accuracy of Laplacian model of incompressible SPH method using higher-order interpolation. Iran J Sci Technol Trans Civ Eng 43:791–805

    Article  Google Scholar 

  • Shobeyri G (2022) Accuracy analysis of improved SPH methods for solution of quadratic PDEs representing potential flows and elasticity problems. J Braz Soc Mech Sci Eng 44(4):146

    Article  Google Scholar 

  • Shobeyri G (2023a) Mixed smoothed particle hydrodynamics method for planar elasticity problems. Iran J Sci Technol Trans Civ Eng 47(1):491–504

    Article  Google Scholar 

  • Shobeyri G (2023b) Improved MPS gradient models for elasticity problems. Iran J Sci Technol Trans Civ Eng 47(3):1831–1843

    Article  Google Scholar 

  • Shobeyri G (2023c) Simulating free surface flows using a new incompressible SPH model improved by MPS method. Arab J Sci Eng 1–14

  • Tamai T, Koshizuka S (2014) Least squares moving particle semi-implicit method. Comput Part Mech 1(3):277–305

    Article  Google Scholar 

  • Tchonkova M, Sture S (2002) A mixed least squares method for solving problems in linear elasticity: theoretical study. Comput Mech 29(4):332–339

    Article  MathSciNet  Google Scholar 

  • Timoshenko SP, Goodier JN (1978) Theory of elasticity, 3rd edn. McGraw Hill, New York

    Google Scholar 

  • Valença J, Mukhandi H, Araújo AG, Couceiro MS, Júlio E (2022) Benchmarking for strain evaluation in CFRP laminates using computer vision: machine learning versus deep learning. Materials 15(18):6310

    Article  Google Scholar 

  • Zhang G, Chen X, Wan D (2019) MPS-FEM coupled method for study of wave-structure interaction. J Mar Sci Appl 18:387–399

    Article  Google Scholar 

  • Zhang Z, Liew KM, Cheng Y (2008) Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng Anal Boundary Elem 32(2):100–107

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Faculty of Civil, Water and Environmental Engineering, Shahid Beheshti University, Tehran, Iran

    Gholamreza Shobeyri, Seyed Hossein Ghoreishi Najafabadi & Mehrdad Abed

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  1. Gholamreza Shobeyri
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  2. Seyed Hossein Ghoreishi Najafabadi
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  3. Mehrdad Abed
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Correspondence to Seyed Hossein Ghoreishi Najafabadi.

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Shobeyri, G., Najafabadi, S.H.G. & Abed, M. A Comparative Study on Two Mixed Least Squares Meshless Models with Improved SPH, MPS and CPM Methods to Solve Elasticity Problems. Iran J Sci Technol Trans Mech Eng (2024). https://doi.org/10.1007/s40997-023-00742-x

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