-
無網格法
(一種計算方法)
鎖定
無網格法發展歷史
有限元(finite element method 簡稱 FEM)在學術和工業已經有了很多的成就
[4]
。但是,有限元也存在很多短板。其中最嚴重的之一就是有限元的計算結果嚴重依賴於網格的劃分質量
[4]
。高質量的有限元網格需要相關專家長時間的參與,尤其是對複雜的集合圖形
[5-6]
。即使使用高質量的網格,有限元模型也經常會在大形變的模擬中失敗
[7]
。
無網格法(meshless methods)的誕生,就是來消除有限元(finite element method)的短板
[4]
。最早的有關無網格法的文獻是由劍橋大學的L.B.Lucy
[8]
和劍橋大學的R.A.Gingold, J.J.Monaghan
[9]
分別於1977年提出。他們介紹了光滑粒子流體動力學(Smoothed particle hydrodynamics,簡稱SPH)
[8-9]
。這兩篇著作的初衷是解決天體物理學(astrophysics)的問題,後來,SPH多用於流體動力學(fluid dynamics)
[10]
。SPH法的計算是基於強形式(strong form),其餘的無網格方法都是在1990年代提出的,他們是基於弱形式(weak form)計算的。最開始主要是用於計算材料力學的
[2]
[4]
。Galerkin法(Element free Galerkin,簡稱EFG)是第一個基於弱形式的無網格法,於1994年被T. Belytschko等人提出。
[11]
無網格法無網格法分類
該法大致可分成兩類:一類是以Lagrange方法為基礎的粒子法(Particle method),如光滑粒子流體動力學(Smoothed particle hydrodynamics,簡稱SPH)法,和在其基礎上發展的運動粒子半隱式(Moving-particle semi-implicit,簡稱MPS)法等;另一類是以Euler方法為基礎的無格子法(Gridless methods),如無格子Euler/N—S算法(Gridless Euler/Navier-Stokes solution algorithm)和無單元Galerkin法(Element free Galerkin,簡稱EFG)等。
無網格方法可以方便地利用座標點計算模擬複雜形狀流場計算,但不足之處是在高雷諾數流動時提高數值計算精度較困難。
無網格方法中比較常見的還有徑向基函數方法(Radious Basis Function),主要使用某徑向基函數(如(MQ)f(r)=r^5)的組合,來逼近原函數。吳忠敏院士在這方面有比較突出的工作。以上方法中,無網格伽遼金法成為目前影響最大,應用最廣的無網格計算方法,現有的 LS-dyna,Abaqus,Radioss等商業軟件都加入了該方法的計算模塊。
- 參考資料
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- 1. G.R. Liu.Meshfree Methods Moving Beyond the Finite Element Method, Second Edition:CRC Press,2009
- 2. T Belytschko.Meshless method : An overview and recent developments:Meshless method : An overview and recent developments,1996
- 3. Lin-jian Wu, Yuan-zhan Wang, Yi Li & Chun-ning Ji.A meshless method by using radial basis function for numerical solutions of wave shoaling equation:Springer Link,2019
- 4. Vinh Phu Nguyen, Timon Rabczuk, St´ephane Bordas, Marc Duflot.Meshless methods: A review and computer implementation aspects:Mathematics and Computers in Simulation,2008
- 5. Yue Yu, George Bourantas, Benjamin Zwick, Grand Joldes, Tina Kapur, Sarah Frisken, Ron Kikinis, Arya Nabavi, Alexandra Golby, Adam Wittek, Karol Miller.Computer simulation of tumour resection-induced brain deformation by a meshless approach:International Journal for Numerical Methods in Biomechanical Engineering,2021
- 6. Horton A, Wittek A, Joldes G, Miller K.A meshless Total Lagrangian explicit dynamics algorithm for surgical simulation:Int J Numer Methods Biomed Eng,2010
- 7. Joldes G, Bourantas G, Zwick B, et al.Suite of meshless algorithms for accurate computation of soft tissue deformation for surgical simulation:Med Image Anal,2019
- 8. L.B. Lucy.A numerical approach to the testing of the fission hypothesis:The Astronomical Journa,1977
- 9. R. A. Gingold, J. J. Monaghan.Smoothed particle hydrodynamics: theory and application to non-spherical stars:Monthly Notices of the Royal Astronomical Society,1977
- 10. J. Bonet, S. Kulasegaram.Correction and stabilization of smooth particle hydrodynamics methods with application in metal forming simulations:Int. J. Numer. Methods Eng,2000
- 11. T. Belytschko, Y.Y. Lu, L. Gu.Element-free Galerkin methods:Int. J. Numer. Methods Eng,1994
- 12. Mehdi Ghommem, George Bourantas, Adam Wittek, Karol Miller, Muhammad R Hajj.Hydrodynamic modeling and performance analysis of bio-inspired swimming:Ocean Engineering,2020
- 13. Saima Safdar, Grand Joldes, Benjamin Zwick, George Bourantas, Ron Kikinis, Adam Wittek, Karol Miller.Automatic Framework for Patient-Specific Biomechanical Computations of Organ Deformation:Computational Biomechanics for Medicine,2021
- 14. Karol Miller, A. Horton, Grand Joldes, Adam Wittek.Beyond finite elements: A comprehensive, patient-specific neurosurgical simulation utilizing a meshless method:Journal of Biomechanics,2012
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