مقایسه دقت یک روش عددی ارتقاء یافته MPS و یک روش بدون شبکه مبتنی بر بسط تیلور و حداقل مربعات

نوع مقاله : مقاله کامل پژوهشی

نویسنده

دانشکده مهندسی عمران، آب و محیط‌زیست، دانشگاه شهید بهشتی، تهران، ایران

10.22034/ceej.2025.64486.2397

چکیده

روش‌های عددی بدون شبکه، در سالیان اخیر برای حل مسائل مختلف مهندسی بسیار استفاده شده‌اند. روش نیمه‌ضمنی ذرات متحرک (MPS) (Moving Particle Semi-implicit) و روش هیدرودینامیک ذرات هموار (SPH) (Smoothed Particle Hydrodynamics) دو نمونه مهم از روش‌های بدون شبکه هستند؛ اما دقت آن‌ها زیاد نیست. اخیراً دو مدل از این روش‌ها با دقت بسیار خوب پیشنهاد شده است که حتی نسبت به بسیاری از مدل‌های موجود ارتقاء‌ یافته، از دقت بالاتری برخوردار می‌باشند. همچنین یک روش عددی مبتنی بر بسط تیلور (Taylor series) و حداقل مربعات نیز در این تحقیق مورد توجه است. ازآنجایی‌که دقت دو روش ارتقاء یافته MPS و SPH تقریباً یکسان است روش MPS انتخاب می‌گردد و دقت آن با روش عددی مبتنی بر بسط تیلور مقایسه می‌شود. برای این منظور چندین معادله دیفرانسیلی بیضوی و جریان‌های پتانسیل که شامل مشتقات مرتبه دوم می‌باشند، به‌عنوان مسائل مرجع انتخاب شده‌اند. برای تمامی این مسائل، روش عددی مبتنی بر بسط تیلور از دقت بالاتری نسبت به‌روش ارتقاء یافته MPS برخوردار می‌باشد. به‌عنوان مثال، روش مبتنی بر بسط تیلور در حل معادله دیفرانسیل سوم این تحقیق، به خطای زیر 5 درصد می‌رسد درحالی‌که خطای روش MPS نزدیک به 20 درصد می‌باشد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Comparison of the Accuracy of an Improved MPS Numerical Method and a Meshless Method Based on Taylor Expansion and Least Squares

نویسنده [English]

  • Gholamreza Shobeyri
Faculty of Civil, Water and Environmental Engineering, Shahid Beheshti University, Tehran, Iran
چکیده [English]

The Moving Particle Semi-implicit (MPS) is one of the efficient meshless numerical methods that has been successfully used to analyze various engineering problems. In this method, the approximation of the derivatives of the function is performed based on a weighted averaging by interpolation functions. Despite its inherent capabilities, this method has serious shortcomings, especially in terms of accuracy. Many efforts have been made to improve the accuracy of this method. In this study, the accuracy of one of the most efficient MPS models (Model 1 in this study) is compared with a meshless method based on the least squares technique and Taylor expansion (Model 2 in this study) for solving elliptic differential equations. Numerical results show that numerical Model 2 produces much better results and under these conditions, it can be expected that this numerical model can be successfully used to solve many engineering problems.

کلیدواژه‌ها [English]

  • Meshless numerical methods
  • Elliptic differential equations
  • MPS method
  • Taylor expansion
  • Least squares

مراجع

Chen X, Cui J, Li MY, “Numerical simulation and energy extraction power fitting of OWSCs under regular waves using SPH method”, Ocean Engineering, 2023, 283, 115077. https://doi.org/10.1016/j.oceaneng.2023.115077
Daneshvar FA, Rakhshandehroo GR, Talebbeydokhti, N, “New modified gradient models for MPS method applied to free-surface flow simulations”, Applied Ocean Research, 2017, 66, 95-116. https://doi.org/10.1016/j.apor.2017.05.009
Duan G, Koshizuka S, Yamaji A, Chen B, Li X, Tamai T, “An accurate and stable multiphase moving particle semi‐implicit method based on a corrective matrix for all particle interaction models”, International Journal for Numerical Methods in Engineering, 2018, 115 (10), 1287-1314. https://doi.org/10.1002/nme.5844
Gingold RA, Monaghan JJ, “Smoothed particle hydrodynamics: theory and application to non-spherical stars”, Monthly Notices of The Royal Astronomical Society, 1977, 181 (3), 375-389. https://doi.org/10.1093/mnras/181.3.375
Gou W, Zhang S, Zheng Y, “Improvement of pressure calculations in the moving particle semi-implicit method for free-surface flows”, International Journal of Computational Methods, 2020, 17 (09), 1950062. https://doi.org/10.1142/S0219876219500622
Han L, Hu X, “SPH modeling of fluid-structure interaction”, Journal of Hydrodynamics, 2018, 30, 62-69. https://doi.org/10.1007/s42241-018-0006-9
Jandaghian M, Shakibaeinia A, “An enhanced weakly-compressible MPS method for free-surface flows”, Computer methods in Applied mechanics and Engineering, 2020, 360, 112771. https://doi.org/10.48550/arXiv.2110.11447
Koh CG, Gao M, Luo C, “A new particle method for simulation of incompressible free surface flow problems”, International Journal for Numerical Methods in Engineering, 2012, 89 (12), 1582-1604. https://doi.org/10.1002/nme.3303
Li D, Zhang H, Qin G, “A modified MPS method with a split-pressure Poisson equation and a virtual particle for simulating free surface flows”, Journal of Marine Science and Engineering, 2023, 11 (1), 215. https://doi.org/10.3390/jmse11010215
Matsunaga T, Koshizuka S, “Stabilized LSMPS method for complex free-surface flow simulation”, Computer Methods in Applied Mechanics and Engineering, 2022, 389, 114416. https://doi.org/10.1016/j.cma.2021.114416
Mitsume N, Yoshimura S, Murotani K, Yamada T, “MPS-FEM partitioned coupling approach for fluid–structure interaction with free surface flow”, International Journal of Computational Methods, 2014, 11 (4), 1350101. https://doi.org/10.1142/S0219876213501016
Pan X, Huai-Xin Z, Xue-Yao S, “Numerical simulation of sloshing with large deforming free surface by MPS-LES method”, China Ocean Engineering, 2012 26 (4), 653-668. https://doi.org/10.1007/s13344-012-0049-6
Peer A, Gissler C, Band S, Teschner M, “An implicit SPH formulation for incompressible linearly elastic solids”, Computer Graphics Forum, 2018, 37 (6), 135-148. https://doi.org/10.1111/cgf.13317
Razavitoosi SL, Ayyoubzadeh SA, Valizadeh A, “Two-phase SPH modelling of waves caused by dam break over a movable bed”, International Journal of Sediment Research, 2014, 29 (3), 344-356. https://doi.org/10.1016/S1001-6279(14)60049-4
Shobeyri G, Madadi HR, “An improvement in MPS method using Voronoi diagram and a new kernel function”, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2018, 40, 1-10. https://doi.org/10.1007/s40430-018-1121-9
Shobeyri G, “Using a modified MPS gradient model to improve accuracy of SPH method for Poisson equations”, Computational Particle Mechanics, 2023, 5, 1113-1126. https://doi.org/10.1007/s40571-022-00549-8
Shobeyri G, “Improved MPS models for simulating free surface flows”, Mathematics and Computers in Simulation, 2024a, 218, 79-97. https://doi.org/10.1016/j.matcom.2023.11.015
Shobeyri G, “Novel SPH and MPS laplacian models improved by mls method for solving poisson equations”, Numerical Methods in Civil Engineering, 2024b, 9 (2), 29-39. https://doi.org/10.61186/NMCE.2406.1061
Shobeyri G, Ghoreishi Najafabadi S, Abed M, “A Comparative Study on Two Mixed Least Squares Meshless Models with Improved SPH, MPS and CPM Methods to Solve Elasticity Problems”, Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 2024, 48 (4), 1565-1580. https://doi.org/10.1007/s40997-023-00742-x
Sun Z, Djidjeli K, Xing JT, “Modified MPS method for the 2D fluid structure interaction problem with free surface”, Computers and Fluids, 2015, 122, 47-65. https://doi.org/10.1016/j.compfluid.2015.08.017
Tan H, Chen S, “A hybrid DEM-SPH model for deformable landslide and its generated surge waves”, Advances in water resources, 2017, 108, 256-276. https://doi.org/10.1016/j.advwatres.2017.07.023
Wu J, Zhang G, Sun Z, Yan H, Zhou B, “An improved MPS method for simulating multiphase flows characterized by high-density ratios and violent deformation of interface”, Computer Methods in Applied Mechanics and Engineering, 2023, 412, 116103. https://doi.org/10.1016/j.cma.2023.116103
Zha R, Peng H, Qiu W, “An improved higher-order moving particle semi-implicit method for simulations of two-dimensional hydroelastic slamming”, Physics of Fluids, 2021, 33 (3). https://doi.org/10.1063/5.0033491
Zhang ZL, Walayat K, Chang JZ, Liu MB, “Meshfree modeling of a fluid‐particle two‐phase flow with an improved SPH method”, International Journal for Numerical Methods in Engineering, 2018, 116 (8), 530-569. https://doi.org/10.1002/nme.5935
Zhang GY, Zhao WW, Wan DC, “Numerical simulations of sloshing waves in vertically excited square tank by improved MPS method”, Journal of Hydrodynamics, 2022, 34 (1), 76-84. https://doi.org/10.1007/s42241-022-0008-5
Zheng X, Ma Q, Shao S, Khayyer A, Zheng X, Ma Q, Shao S, Khayyer A, “Modelling of violent water wave propagation and impact by incompressible SPH with first-order consistent kernel interpolation scheme”, Water, 2017, 9 (6), 400. https://doi.org/10.3390/w9060400
نشریه مهندسی عمران و محیط زیست
دوره 55، شماره 121 - شماره پیاپی 121
زمستان 1404
بهمن 1404
صفحه 27-38

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