Applied Mathematics and Mechanics >
Numerical investigation on a comprehensive high-order finite particle scheme
Received date: 2024-12-24
Revised date: 2025-04-23
Online published: 2025-06-06
Supported by
Project supported by the National Natural Science Foundation of China (No. 12002290)
Copyright
In the field of discretization-based meshfree/meshless methods, the improvements in the higher-order consistency, stability, and computational efficiency are of great concerns in computational science and numerical solutions to partial differential equations. Various alternative numerical methods of the finite particle method (FPM) frame have been extended from mathematical theories to numerical applications separately. As a comprehensive numerical scheme, this study suggests a unified resolved program for numerically investigating their accuracy, stability, consistency, computational efficiency, and practical applicability in industrial engineering contexts. The high-order finite particle method (HFPM) and corrected methods based on the multivariate Taylor series expansion are constructed and analyzed to investigate the whole applicability in different benchmarks of computational fluid dynamics. Specifically, four benchmarks are designed purposefully from statical exact solutions to multifaceted hydrodynamic tests, which possess different numerical performances on the particle consistency, numerical discretized forms, particle distributions, and transient time evolutional stabilities. This study offers a numerical reference for the current unified resolved program.
Yudong LI, Yan LI, Chunfa WANG, P. JOLI, Zhiqiang FENG . Numerical investigation on a comprehensive high-order finite particle scheme[J]. Applied Mathematics and Mechanics, 2025 , 46(6) : 1187 -1214 . DOI: 10.1007/s10483-025-3262-9
| [1] | BELYTSCHKO, T., KRONGAUZ, Y., ORGAN, D., FLEMING, M., and KRYSL, P. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 139(1-4), 3–47 (1996) |
| [2] | ANTUONO, M., COLAGROSSI, A., MARRONE, S., and LUGNI, C. Propagation of gravity waves through an SPH scheme with numerical diffusive terms. Computer Physics Communications, 182(4), 866–877 (2011) |
| [3] | CHENG, Y. J., LI, Y., TAO, L., JOLI, P., and FENG, Z. Q. An adaptive smoothed particle hydrodynamics for metal cutting simulation. Acta Mechanica Sinica, 38(10), 422186 (2022) |
| [4] | ZHANG, Z. L. and LIU, M. B. Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two- and three-dimensional spaces. Engineering Analysis with Boundary Elements, 83, 141–157 (2017) |
| [5] | XIANG, H. Y. and CHEN, X. W. Numerical study on breakup of DebriSat under hypervelocity impact. Acta Astronautica, 217, 62–74 (2024) |
| [6] | MONAGHAN, J. J. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics, 30, 543–574 (1992) |
| [7] | YE, T., PAN, D. Y., HUANG, C., and LIU, M. B. Smoothed particle hydrodynamics (SPH) for complex fluid flows: recent developments in methodology and applications. Physics of Fluids, 31(1), 011301 (2019) |
| [8] | SHADLOO, M. S., OGER, G., and TOUZé, D. Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: motivations, current state, and challenges. Computers and Fluids, 136, 11–34 (2016) |
| [9] | XU, F., WANG, J. Y., YANG, Y., WANG, L., DAI, Z., and HAN, R. Q. On methodology and application of smoothed particle hydrodynamics in fluid, solid and biomechanics. Acta Mechanica Sinica, 39(2), 722185 (2023) |
| [10] | SIGALOTTI, L. D. G., RENDóN, O., KLAPP, J., VARGAS, C. A., and CRUZ, F. A new insight into the consistency of the SPH interpolation formula. Applied Mathematics and Computation, 356, 50–73 (2019) |
| [11] | FRANCOMANO, E. and PALIAGA, M. Highlighting numerical insights of an efficient SPH method. Applied Mathematics and Computation, 339, 899–915 (2018) |
| [12] | ANTONELLI, L., FRANCOMANO, E., and GREGORETTI, F. A CUDA-based implementation of an improved SPH method on GPU. Applied Mathematics and Computation, 409, 125482 (2021) |
| [13] | PENG, P. P. and CHENG, Y. M. Analyzing three-dimensional transient heat conduction problems with the dimension splitting reproducing kernel particle method. Engineering Analysis with Boundary Elements, 121, 180–191 (2020) |
| [14] | LI, S. F., QIAN, D., LIU, W. K., and BELYTSCHKO, T. A meshfree contact-detection algorithm. Computer Methods in Applied Mechanics and Engineering, 190(24-25), 3271–3292 (2001) |
| [15] | LI, Y. D., FU, Z. J., and TANG, Z. C. Generalized finite difference method for bending and modal analysis of functionally graded carbon nanotube-reinforced composite plates. Chinese Journal of Theoretical and Applied Mechanics, 54(2), 414–424 (2022) |
| [16] | LI, P. W., FAN, C. M., YU, Y. Z., and SONG, L. N. A meshless generalized finite difference scheme for the stream function formulation of the Navier-Stokes equations. Engineering Analysis with Boundary Elements, 152, 154–168 (2023) |
| [17] | FU, Z. J., XIE, Z. Y., JI, S. Y., TSAI, C. C., and LI, A. L. Meshless generalized finite difference method for water wave interactions with multiple-bottom-seated-cylinder-array structures. Ocean Engineering, 195, 106736 (2020) |
| [18] | CHEN, J. K., BERAUN, J. E., and CARNEY, T. C. A corrective smoothed particle method for boundary value problems in heat conduction. International Journal for Numerical Methods in Engineering, 46(2), 231–252 (1999) |
| [19] | LIU, M. B., XIE, W. P., and LIU, G. R. Modeling incompressible flows using a finite particle method. Applied Mathematical Modelling, 29(12), 1252–1270 (2005) |
| [20] | O?ATE, E., IDELSOHN, S., ZIENKIEWICZ, O. C., and TAYLOR, R. L. A finite point method in computational mechanics: applications to convective transport and fluid flow. International Journal for Numerical Methods in Engineering, 39(22), 3839–3866 (1996) |
| [21] | TIWARI, S. and KUHNERT, J. Modeling of two-phase flows with surface tension by finite pointset method (FPM). Journal of Computational and Applied Mathematics, 203(2), 376–386 (2007) |
| [22] | SAUCEDO-ZENDEJO, F. R. and RESéNDIZ-FLORES, E. O. Meshfree numerical approach based on the finite pointset method for static linear elasticity problems. Computer Methods in Applied Mechanics and Engineering, 372, 113367 (2020) |
| [23] | SAUCEDO-ZENDEJO, F. R., MEDRANO-MENDIETA, J. L., and NU?EZ-BRIONES, A. G. A GFDM approach based on the finite pointset method for two-dimensional piezoelectric problems. Engineering Analysis with Boundary Elements, 163, 12–22 (2024) |
| [24] | SAUCEDO-ZENDEJO, F. R. A novel meshfree approach based on the finite pointset method for linear elasticity problems. Engineering Analysis with Boundary Elements, 136, 172–185 (2022) |
| [25] | SAUCEDO-ZENDEJO, F. R. and RESéNDIZ-FLORES, E. O. A new approach for the numerical simulation of free surface incompressible flows using a meshfree method. Computer Methods in Applied Mechanics and Engineering, 324, 619–639 (2017) |
| [26] | RESéNDIZ-FLORES, E. O. and SAUCEDO-ZENDEJO, F. R. Meshfree numerical simulation of free surface thermal flows in mould filling processes using the finite pointset method. International Journal of Thermal Sciences, 127, 29–40 (2018) |
| [27] | JIANG, T., REN, J. L., YUAN, J. Y., ZHOU, W., and WANG, D. S. A least-squares particle model with other techniques for 2D viscoelastic fluid/free surface flow. Journal of Computational Physics, 407, 109255 (2020) |
| [28] | YANG, Y., XU, F., ZHANG, M., and WANG, L. An effective improved algorithm for finite particle method. International Journal of Computational Methods, 13(4), 1641009 (2016) |
| [29] | WANG, L., XU, F., and YANG, Y. An improved specified finite particle method and its application to transient heat conduction. International Journal of Computational Methods, 14(5), 1750050 (2017) |
| [30] | BONET, J. and LOK, T. Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. Computer Methods in Applied Mechanics and Engineering, 180(1-2), 97–115 (1999) |
| [31] | HUANG, C., LEI, J. M., LIU, M. B., and PENG, X. Y. A kernel gradient free (KGF) SPH method. International Journal for Numerical Methods in Fluids, 78(11), 691–707 (2015) |
| [32] | ZHANG, Z. L. and LIU, M. B. A decoupled finite particle method for modeling incompressible flows with free surfaces. Applied Mathematical Modelling, 60, 606–633 (2018) |
| [33] | NASAR, A. M. A., FOURTAKAS, G., LIND, S. J., KING, J. R. C., ROGERS, B. D., and STANSBY, P. K. High-order consistent SPH with the pressure projection method in 2-D and 3-D. Journal of Computational Physics, 444, 110563 (2021) |
| [34] | LIU, M. B. and LIU, G. R. Smoothed particle hydrodynamics (SPH): an overview and recent developments. Archives of Computational Methods in Engineering, 17, 25–76 (2010) |
| [35] | LI, Y. D., LI, Y., and FENG, Z. Q. A coupled particle model with particle shifting technology for simulating transient viscoelastic fluid flow with free surface. Journal of Computational Physics, 488, 112213 (2023) |
| [36] | LIU, M. B. and LIU, G. R. Restoring particle consistency in smoothed particle hydrodynamics. Applied Numerical Mathematics, 56(1), 19–36 (2006) |
| [37] | LIU, M. B., LIU, G. R., and LAM, K. Y. Constructing smoothing functions in smoothed particle hydrodynamics with applications. Journal of Computational and Applied Mathematics, 155(2), 263–284 (2003) |
| [38] | CHERNIH, A. and HUBBERT, S. Closed form representations and properties of the generalised Wendland functions. Journal of Approximation Theory, 177, 17–33 (2014) |
| [39] | FATEHI, R. and MANZARI, M. T. Error estimation in smoothed particle hydrodynamics and a new scheme for second derivatives. Computers and Mathematics with Applications, 61(2), 482–498 (2011) |
| [40] | MONAGHAN, J. J. and RAFIEE, A. A simple SPH algorithm for multi-fluid flow with high density ratios. International Journal for Numerical Methods in Fluids, 71(5), 537–561 (2013) |
| [41] | LI, Y. D., LI, Y., JOLI, P., CHEN, H. J., and FENG, Z. Q. Extension of finite particle method simulating thermal-viscoelastic flow and fluid-rigid body interactional process in weakly compressible smoothed particle hydrodynamics scheme. Physics of Fluids, 36(4), 043307 (2024) |
| [42] | LI, Y. D., LI, Y., and FENG, Z. Q. Extension of decoupled finite particle method to simulate non-isothermal free surface flow. International Journal of Multiphase Flow, 167, 104532 (2023) |
| [43] | SIGALOTTI, L. D. G., KLAPP, J., RENDóN, O., VARGAS, C. A., and PE?A-POLO, F. On the kernel and particle consistency in smoothed particle hydrodynamics. Applied Numerical Mathematics, 108, 242–255 (2016) |
| [44] | XU, R., STANSBY, P. K., and LAURENCE, D. Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. Journal of Computational Physics, 228(18), 6703–6725 (2009) |
| [45] | LIND, S. J., XU, R., STANSBY, P. K., and ROGERS, B. D. Incompressible smoothed particle hydrodynamics for free-surface flows: a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. Journal of Computational Physics, 231(4), 1499–1523 (2012) |
| [46] | LIU, Z., WEI, G. F., and WANG, Z. M. The radial basis reproducing kernel particle method for geometrically nonlinear problem of functionally graded materials. Applied Mathematical Modelling, 85, 244–272 (2020) |
| [47] | FLYER, N., BARNETT, G. A., and WICKER, L. J. Enhancing finite differences with radial basis functions: experiments on the Navier-Stokes equations. Journal of Computational Physics, 316, 39–62 (2016) |
| [48] | VOROZHTSOV, E. V. and KISELEV, S. P. Higher-order symplectic integration techniques for molecular dynamics problems. Journal of Computational Physics, 452, 110905 (2022) |
| [49] | KOTYCZKA, P. and LEFEVRE, L. Discrete-time port-Hamiltonian systems: a definition based on symplectic integration. Systems and Control Letters, 133, 104530 (2019) |
/
| 〈 |
|
〉 |
Email Alert
RSS