G space

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G space [1] is a functional space used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solve PDEs in particular solid mechanics as well as fluid dynamics problems.

Description

For simplicity we choose elasticity problems for our discussion. In a weak formulation (such as in the FEM), displacement functions need to be in a proper Hilbert space, meaning that we need to make sure that the assumed displacement function is continuous over the entire problems domain. In a discrete setting (e.g., FEM), we construct the function using elements, but have to make sure that it is continuous along all element interfaces. This is also known as compatibility conditions. To ensure the compatibility, however, care must be taken, and FEM techniques [2] should apply.

The G space theory accommodates functions that may be discontinuous. This is done by using the so-called generalized gradient smoothing technique,[3] with which one can approximate the gradient of displacement functions in a proper G space.[4] Since we do not have to actually perform even the 1st differentiation to the assumed displacement functions, the requirements on the consistence of the functions are further reduced, and hence the Weakened weak form or W2 form can be used to create stable and convergent computational methods. The stability is ensured by the so-called positivity conditions, and the convergence to the exact solution is ensure the admissible conditions on the assumed gradient (strain) fields.[5]

History

The development of G space theory started from the works on meshfree methods.[6][7] The G Space theory forms the foundation for the W2 formulations, leading to various W2 models. The W2 models work well with triangular meshes and are insensitive to mesh distortion. Because triangular meshes can be generated automatically, the model becomes much easier in re-meshing and hence automation in modeling and simulation. In addition, W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the Smoothed Point Interpolation Methods (or S-PIM).[8] The S-PIM can be node-based (known as NS-PIM or LC-PIM),[9] edge-based (ES-PIM),[10] and cell-based (CS-PIM).[11] The NS-PIM was developed using the so-called SCNI technique.[12] It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.[13] The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further room for future developments.

The S-FEM is largely the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler. It also has variations of Node-based Smoothed FEM (NS-FEM) [14] Edge-based Smoothed FEM (NS-FEM),[15] Face-based Smoothed FEM (NS-FEM),[16] Cell-based Smoothed FEM (NS-FEM),[17][18][19] Edge/node-based Smoothed FEM (NS/ES-FEM),[20] as well as Alpha FEM method (Alpha FEM).[21][22]

Applications

Numerical methods built on G space theory have been applied to solve the following physical problems:

See also

References

  1. ↑ Liu GR, "On G Space Theory", International Journal of Computational Methods, Vol. 6 Issue: 2,257-289, 2009
  2. ↑ Liu GR, The Finite element method – a practical course”, Elsevier (BH) ISBN 0124529518, 2003
  3. ↑ Liu GR, "A Generalized Gradient Smoothing Technique and the Smoothed Bilinear Form for Galerkin Formulation of a Wide Class of Computational Methods", International Journal of Computational Methods Vol.5 Issue: 2, 199–236, 2008
  4. ↑ Liu GR, "On G Space Theory", International Journal of Computational Methods, Vol. 6 Issue: 2,257-289, 2009
  5. ↑ Liu GR, "On G Space Theory", International Journal of Computational Methods, Vol. 6 Issue: 2,257-289, 2009
  6. ↑ Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
  7. ↑ G.R. Liu. "A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory and Part II applications to solid mechanics problems." International Journal for Numerical Methods in Engineering, 81: 1093-1126, 2010
  8. ↑ Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
  9. ↑ Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY and Han X, "A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems", International Journal of Computational Methods, 2(4): 645-665, 2005.
  10. ↑ G.R. Liu, G.R. Zhang. "Edge-based Smoothed Point Interpolation Methods". International Journal of Computational Methods, 5(4): 621-646, 2008
  11. ↑ G.R. Liu, G.R. Zhang. "A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method." International Journal of Computational Methods, 6(1): 147-179, 2009
  12. ↑ Chen, J. S., Wu, C. T., Yoon, S. and You, Y. (2001). "A stabilized conforming nodal integration for Galerkin mesh-free methods". International Journal of Numerical Methods. Eng. 50: 435–466.
  13. ↑ G. R. Liu and G. Y. Zhang. "Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM)." International Journal for Numerical Methods in Engineering, 74: 1128-1161, 2008.
  14. ↑ Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY (2009) A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Computers and Structures; 87: 14-26.
  15. ↑ Liu GR, Nguyen-Thoi T, Lam KY (2009) An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses in solids. Journal of Sound and Vibration; 320: 1100-1130.
  16. ↑ Nguyen-Thoi T, Liu GR, Lam KY, GY Zhang (2009) "A Face-based Smoothed Finite Element Method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements." International Journal for Numerical Methods in Engineering; 78: 324-353
  17. ↑ Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed finite element method for mechanics problems. Computational Mechanics; 39: 859-877
  18. ↑ Dai KY, Liu GR (2007) Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration; 301: 803-820.
  19. ↑ Dai KY, Liu GR, Nguyen-Thoi T (2007) An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elements in Analysis and Design; 43: 847－860.
  20. ↑ Li Y, Liu GR, Zhang GY, "An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements", Finite Elements in Analysis and Design Vol.47 Issue: 3, 256-275, 2011
  21. ↑ Liu GR, Nguyen-Thoi T, Lam KY (2009) A novel FEM by scaling the gradient of strains with factor α (αFEM). Computational Mechanics; 43: 369-391
  22. ↑ Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, Xu X (2009) A novel weak form and a superconvergent alpha finite element method (SαFEM) for mechanics problems using triangular meshes. Journal of Computational Physics; 228: 4055-4087
  23. ↑ Cui XY, Liu GR, Li GY, et al. A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells, International Journal for Numerical Methods in Engineering Vol.85 Issue: 8 , 958-986, 2011
  24. ↑ Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates, International Journal for Numerical Methods in Engineering Vol. 84 Issue: 10, 1222-1256, 2010
  25. ↑ Liu GR, Nourbakhshnia N, Zhang YW, "A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems", Engineering Fracture Mechanics Vol.78 Issue: 6 Pages: 863-876, 2011
  26. ↑ Liu GR, Chen L, Nguyen-Thoi T, et al. A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems, International Journal for Numerical Methods in Engineering Vol.83 Issue: 11, 1466-1497, 2010
  27. ↑ Liu GR, Nourbakhshnia N, Chen L, et al. "A Novel General Formulation for Singular Stress Field Using the Es-Fem Method for the Analysis of Mixed-Mode Cracks", International Journal of Computational Methods Vol. 7 Issue: 1, 191-214, 2010
  28. ↑ Zhang ZB, Wu SC, Liu GR, et al. Nonlinear Transient Heat Transfer Problems using the Meshfree ES-PIM, International Journal of Nonlinear Sciences and Numerical Simulation Vol.11 Issue: 12, 1077-1091, 2010
  29. ↑ Wu SC, Liu GR, Cui XY, et al. An edge-based smoothed point interpolation method (ES-PIM) for heat transfer analysis of rapid manufacturing system, International Journal of Heat and Mass Transfer Vol.53 Issue: 9-10, 1938-1950, 2010
  30. ↑ He ZC, Cheng AG, Zhang GY, et al. Dispersion error reduction for acoustic problems using the edge-based smoothed finite element method (ES-FEM), International Journal for Numerical Methods in Engineering Vol. 86 Issue: 11 Pages: 1322-1338, 2011
  31. ↑ He ZC, Liu GR, Zhong ZH, et al. "A coupled ES-FEM/BEM method for fluid-structure interaction problems", Engineering Analysis with Boundary Elements Vol. 35 Issue: 1, 140-147, 2011
  32. ↑ Zhang ZQ, Liu GR, Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods, International Journal for Numerical Methods in Engineering Vol.84 Issue: 2,149-178, 2010
  33. ↑ Zhang ZQ, Liu GR, An edge-based smoothed finite element method (ES-FEM) using 3-node triangular elements for 3D non-linear analysis of spatial membrane structures, International Journal for Numerical Methods in Engineering, Vol. 86 Issue: 2 135-154, 2011
  34. ↑ Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, et al. Adaptive analysis using the node-based smoothed finite element method (NS-FEM), International Journal for Numerical Methods in Biomedical Engineering Vol. 27 Issue: 2, 198-218, 2011
  35. ↑ Li Y, Liu GR, Zhang GY, "An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements", Finite Elements in Analysis and Design Vol.47 Issue: 3, 256-275, 2011
  36. ↑ Li E, Liu GR, Tan V, et al. An efficient algorithm for phase change problem in tumor treatment using alpha FEM, International Journal of Thermal Sciences Vol.49 Issue: 10, 1954-1967, 2010
  37. ↑ Tran TN, Liu GR, Nguyen-Xuan H, et al. An edge-based smoothed finite element method for primal-dual shakedown analysis of structures, International Journal for Numerical Methods in Engineering Vol.82 Issue: 7, 917-938, 2010
  38. ↑ Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9

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