Recent developments in T-Trefftz and F-Trefftz Finite Element Methods

Authors

  • Yi Xiao  Research School of Engineering, Australian National University, Acton, ACT 2601, Australia

Keywords:

Finite Element Method, Trefftz Method, Fundamental Solution, Variational functional

Abstract

This paper presents an overview of both T-Trefftz and F-Trefftz finite element methods (FEM) and its application in various engineering problems. Recent developments on the T-Trefftz finite element formulation of nonlinear problems of minimal surface, F-Trefftz methods for composite, skin tissue, and functionally graded materials are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions or fundamental solutions. Generation of elemental stiffness equations from the modified variational principle is also discussed. Finally, a brief summary of the approach is provided and future trends in this field are identified.

References

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  12. L. Cao, H. Wang, Q.H. Qin, Fundamental solution based graded element model for steady-state heat transfer in FGM, Acta Mechanica Solida Sinica, 25 (2012) 377-392.
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  20. Q.H. Qin, Formulation of hybrid Trefftz finite element method for elastoplasticity, Applied mathematical modelling, 29 (2005) 235-252.
  21. Q.H. Qin, Trefftz plane elements of elastoplasticity with p-extension capabilities, Journal of Mechanical Engineering, 56 (2005) 40-59.
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  36. Q.H. Qin, Nonlinear analysis of thick plates by HT FE approach, Computers & structures, 61 (1996) 271-281.
  37. Q.H. Qin, S. Diao, Nonlinear analysis of thick plates on an elastic foundation by HT FE with p-extension capabilities, International Journal of Solids and Structures, 33 (1996) 4583-4604.
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  39. Q.H. Qin, Variational formulations for TFEM of piezoelectricity, International Journal of Solids and Structures, 40 (2003) 6335-6346.
  40. Q.H. Qin, Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach, Computational Mechanics, 31 (2003) 461-468.
  41. Q.H. Qin, Mode III fracture analysis of piezoelectric materials by Trefftz BEM, Structural Engineering and Mechanics, 20 (2005) 225-240.
  42. Q.H. Qin, K.Y. Wang, Application of hybrid-Trefftz finite element method fractional contact problems, Computer Assisted Mechanics and Engineering Sciences, 15 (2008) 319-336.
  43. K. Wang, Q.H. Qin, Y. Kang, J. Wang, C. Qu, A direct constraint?Trefftz FEM for analysing elastic contact problems, International journal for numerical methods in engineering, 63 (2005) 1694-1718.
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  47. H. Wang, Q.H. Qin, Special fiber elements for thermal analysis of fiber-reinforced composites, Engineering Computations, 28 (2011) 1079-1097.
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  49. C. Cao, A. Yu, Q.H. Qin, A new hybrid finite element approach for plane piezoelectricity with defects, Acta Mechanica, 224 (2013) 41-61.
  50. H. Wang, Q.H. Qin, Fracture analysis in plane piezoelectric media using hybrid finite element model, in:  The 13th International Conference on Fracture, 2013.
  51. C. Cao, Q.H. Qin, A. Yu, A new hybrid finite element approach for three-dimensional elastic problems, Archives of Mechanics, 64 (2012) 261–292.
  52. H. Wang, L. Cao, Q.H. Qin, Hybrid Graded Element Model for Nonlinear Functionally Graded Materials, Mechanics of Advanced Materials and Structures, 19 (2012) 590-602.
  53. H. Wang, Q.H. Qin, Boundary Integral Based Graded Element For Elastic Analysis of 2D Functionally Graded Plates, European Journal of Mechanics-A/Solids, 33 (2012) 12-23.
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  58. H. Wang, Q.H. Qin, Implementation of fundamental-solution based hybrid finite element model for elastic circular inclusions, in:  The Asia-Pacific Congress for Computational Mechanics, 2013.
  59. H. Wang, Q.H. Qin, W.-A. Yao, Improving accuracy of opening-mode stress intensity factor in two-dimensional media using fundamental solution based finite element model, Australian Journal of Mechanical Engineering, 10 (2012) 41-52.
  60. Z.-W. Zhang, H. Wang, Q.H. Qin, Transient Bioheat Simulation of the Laser-Tissue Interaction in Human Skin Using Hybrid Finite Element Formulation, MCB: Molecular & Cellular Biomechanics, 9 (2012) 31-54.
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  1. H.C. Martin, G.F. Carey, Introduction to Finite Element Analysis: Theory and Applications, McGraw-Hill Book Company, New York 1973.
  2. J.H. Argyris, M. Kleiber, Incremental formulation in nonlinear mechanics and large strain elasto-plasticity - natural approach .1, Computer Methods in Applied Mechanics and Engineering, 11 (1977) 215-247.
  3. J.H. Argyris, J.S. Doltsinis, M. Kleiber, Incremental formulation in non-linear mechanics and large strain elasto-plasticity - Natural approach .2, Computer Methods in Applied Mechanics and Engineering, 14 (1978) 259-294.
  4. J. Jirousek, N. Leon, Powerful finite-element for plate bending, Computer Methods in Applied Mechanics and Engineering, 12 (1977) 77-96.
  5. Q.H. Qin, C.X. Mao, Coupled torsional-flexural vibration of shaft systems in mechanical engineering .I. Finite element model, Computers & Structures, 58 (1996) 835-843.
  6. Q.H. Qin, The Trefftz finite and boundary element method, WIT Press, Southampton, 2000.
  7. H. Wang, Q.H. Qin, Hybrid FEM with fundamental solutions as trial functions for heat conduction simulation, Acta Mechanica Solida Sinica, 22 (2009) 487-498.
  8. J.H. Argyris, P.C. Dunne, M. Haase, J. Orkisz, Higher-order simplex elements for large strain analysis - Natural approach, Computer Methods in Applied Mechanics and Engineering, 16 (1978) 369-403.
  9. C.X. Mao, Q.H. Qin, Coupled torsional-flexural vibration of shaft systems in mechanical engineering—II. FE-TM impedance coupling method, Computers & Structures, 58 (1996) 845-849.
  10. Q.H. Qin, Y. Mai, BEM for crack-hole problems in thermopiezoelectric materials, Engineering Fracture Mechanics, 69 (2002) 577-588.
  11. C. Cao, A. Yu, Q.H. Qin, Evaluation of effective thermal conductivity of fiber-reinforced composites by boundary integral based finite element method, International Journal of Architecture, Engineering and Construction, 1 (2012) 14-29.
  12. L. Cao, H. Wang, Q.H. Qin, Fundamental solution based graded element model for steady-state heat transfer in FGM, Acta Mechanica Solida Sinica, 25 (2012) 377-392.
  13. J. Jirousek, Basis for development of large finite-elements locally satisfying all field equations, Computer Methods in Applied Mechanics and Engineering, 14 (1978) 65-92.
  14. Q.H. Qin, H. Wang, Matlab and C programming for Trefftz finite element methods, New York: CRC Press, 2008.
  15. Q.H. Qin, Trefftz finite element method and its applications, Applied Mechanics Reviews, 58 (2005) 316-337.
  16. W. Chen, Z.-J. Fu, Q.H. Qin, Boundary particle method with high-order Trefftz functions, Computers, Materials & Continua (CMC), 13 (2010) 201-217.
  17. H. Wang, Q.H. Qin, D. Arounsavat, Application of hybrid Trefftz finite element method to non?linear problems of minimal surface, International Journal for Numerical Methods in Engineering, 69 (2007) 1262-1277.
  18. H. Wang, Q.H. Qin, X.P. Liang, Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model, Engineering Analysis with Boundary Elements, 36 (2012) 39-46.
  19. Q.H. Qin, Dual variational formulation for Trefftz finite element method of elastic materials, Mechanics Research Communications, 31 (2004) 321-330.
  20. Q.H. Qin, Formulation of hybrid Trefftz finite element method for elastoplasticity, Applied mathematical modelling, 29 (2005) 235-252.
  21. Q.H. Qin, Trefftz plane elements of elastoplasticity with p-extension capabilities, Journal of Mechanical Engineering, 56 (2005) 40-59.
  22. Y. Cui, Q.H. Qin, J.-S. WANG, Application of HT finite element method to multiple crack problems of Mode I, II and III, Chinese Journal of Engineering Mechanics, 23 (2006) 104-110.
  23. Y. Cui, J. Wang, M. Dhanasekar, Q.H. Qin, Mode III fracture analysis by Trefftz boundary element method, Acta Mechanica Sinica, 23 (2007) 173-181.
  24. C. Cao, Q.H. Qin, A. Yu, Micromechanical Analysis of Heterogeneous Composites using Hybrid Trefftz FEM and Hybrid Fundamental Solution Based FEM, Journal of Mechanics, 29 (2013) 661-674.
  25. M. Dhanasekar, J. Han, Q.H. Qin, A hybrid-Trefftz element containing an elliptic hole, Finite Elements in Analysis and Design, 42 (2006) 1314-1323.
  26. Q.H. Qin, X.Q. He, Special elliptic hole elements of Trefftz FEM in stress concentration analysis, Journal of Mechanics and MEMS, 1 (2009) 335-348.
  27. Z.J. Fu, Q.H. Qin, W. Chen, Hybrid-Trefftz finite element method for heat conduction in nonlinear functionally graded materials, Engineering Computations, 28 (2011) 578-599.
  28. J. Jirousek, Q.H. Qin, Application of hybrid-Trefftz element approach to transient heat conduction analysis, Computers & Structures, 58 (1996) 195-201.
  29. Q.H. Qin, Hybrid Trefftz finite-element approach for plate bending on an elastic foundation, Applied Mathematical Modelling, 18 (1994) 334-339.
  30. Q.H. Qin, Postbuckling analysis of thin plates by a hybrid Trefftz finite element method, Computer Methods in Applied Mechanics and Engineering, 128 (1995) 123-136.
  31. Q.H. Qin, Transient plate bending analysis by hybrid Trefftz element approach, Communications in Numerical Methods in Engineering, 12 (1996) 609-616.
  32. Q.H. Qin, Postbuckling analysis of thin plates on an elastic foundation by HT FE approach, Applied Mathematical Modelling, 21 (1997) 547-556.
  33. F. Jin, Q.H. Qin, A variational principle and hybrid Trefftz finite element for the analysis of Reissner plates, Computers & structures, 56 (1995) 697-701.
  34. J. Jirousek, A. Wroblewski, Q.H. Qin, X. He, A family of quadrilateral hybrid-Trefftz p-elements for thick plate analysis, Computer Methods in Applied Mechanics and Engineering, 127 (1995) 315-344.
  35. Q.H. Qin, Hybrid-Trefftz finite element method for Reissner plates on an elastic foundation, Computer Methods in Applied Mechanics and Engineering, 122 (1995) 379-392.
  36. Q.H. Qin, Nonlinear analysis of thick plates by HT FE approach, Computers & structures, 61 (1996) 271-281.
  37. Q.H. Qin, S. Diao, Nonlinear analysis of thick plates on an elastic foundation by HT FE with p-extension capabilities, International Journal of Solids and Structures, 33 (1996) 4583-4604.
  38. C.Y. Lee, Q.H. Qin, H. Wang, Trefftz functions and application to 3D elasticity, Computer Assisted Mechanics and Engineering Sciences, 15 (2008) 251-263.
  39. Q.H. Qin, Variational formulations for TFEM of piezoelectricity, International Journal of Solids and Structures, 40 (2003) 6335-6346.
  40. Q.H. Qin, Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach, Computational Mechanics, 31 (2003) 461-468.
  41. Q.H. Qin, Mode III fracture analysis of piezoelectric materials by Trefftz BEM, Structural Engineering and Mechanics, 20 (2005) 225-240.
  42. Q.H. Qin, K.Y. Wang, Application of hybrid-Trefftz finite element method fractional contact problems, Computer Assisted Mechanics and Engineering Sciences, 15 (2008) 319-336.
  43. K. Wang, Q.H. Qin, Y. Kang, J. Wang, C. Qu, A direct constraint?Trefftz FEM for analysing elastic contact problems, International journal for numerical methods in engineering, 63 (2005) 1694-1718.
  44. H. Wang, Q.H. Qin, Fundamental-solution-based finite element model for plane orthotropic elastic bodies, European Journal of Mechanics-A/Solids, 29 (2010) 801-809.
  45. H. Wang, Q.H. Qin, Fundamental solution-based hybrid finite element analysis for non-linear minimal surface problems, Recent Developments in Boundary Element Methods: A Volume to Honour Professor John T. Katsikadelis, (2010) 309.
  46. H. Wang, Q.H. Qin, Numerical implementation of local effects due to two-dimensional discontinuous loads using special elements based on boundary integrals, Engineering Analysis with Boundary Elements, 36 (2012) 1733-1745.
  47. H. Wang, Q.H. Qin, Special fiber elements for thermal analysis of fiber-reinforced composites, Engineering Computations, 28 (2011) 1079-1097.
  48. C. Cao, Q.H. Qin, A. Yu, Hybrid fundamental-solution-based FEM for piezoelectric materials, Computational Mechanics, 50 (2012) 397-412.
  49. C. Cao, A. Yu, Q.H. Qin, A new hybrid finite element approach for plane piezoelectricity with defects, Acta Mechanica, 224 (2013) 41-61.
  50. H. Wang, Q.H. Qin, Fracture analysis in plane piezoelectric media using hybrid finite element model, in:  The 13th International Conference on Fracture, 2013.
  51. C. Cao, Q.H. Qin, A. Yu, A new hybrid finite element approach for three-dimensional elastic problems, Archives of Mechanics, 64 (2012) 261–292.
  52. H. Wang, L. Cao, Q.H. Qin, Hybrid Graded Element Model for Nonlinear Functionally Graded Materials, Mechanics of Advanced Materials and Structures, 19 (2012) 590-602.
  53. H. Wang, Q.H. Qin, Boundary Integral Based Graded Element For Elastic Analysis of 2D Functionally Graded Plates, European Journal of Mechanics-A/Solids, 33 (2012) 12-23.
  54. H. Wang, Q.H. Qin, FE approach with Green’s function as internal trial function for simulating bioheat transfer in the human eye, Archives of Mechanics, 62 (2010) 493-510.
  55. H. Wang, Q.H. Qin, Computational bioheat modeling in human eye with local blood perfusion effect, in:  Human Eye Imaging and Modeling, CRC Press, 2012, pp. 311-328.
  56. H. Wang, Q.H. Qin, A fundamental solution based FE model for thermal analysis of nanocomposites, Boundary elements and other mesh Reduction methods XXXIII', 33rd International Conference on Boundary Elements and other Mesh Reduction Methods, ed. CA Brebbia and V. Popov, WIT Press, UK, (2011) 191-202.
  57. H. Wang, Q.H. Qin, A new special element for stress concentration analysis of a plate with elliptical holes, Acta Mechanica, 223 (2012) 1323-1340.
  58. H. Wang, Q.H. Qin, Implementation of fundamental-solution based hybrid finite element model for elastic circular inclusions, in:  The Asia-Pacific Congress for Computational Mechanics, 2013.
  59. H. Wang, Q.H. Qin, W.-A. Yao, Improving accuracy of opening-mode stress intensity factor in two-dimensional media using fundamental solution based finite element model, Australian Journal of Mechanical Engineering, 10 (2012) 41-52.
  60. Z.-W. Zhang, H. Wang, Q.H. Qin, Transient Bioheat Simulation of the Laser-Tissue Interaction in Human Skin Using Hybrid Finite Element Formulation, MCB: Molecular & Cellular Biomechanics, 9 (2012) 31-54.
  61. H. Wang, Q.H. Qin, A fundamental solution-based finite element model for analyzing multi-layer skin burn injury, Journal of Mechanics in Medicine and Biology, 12 (2012) 1250027.
  62. C.B. Morrey, Multiple integrals in the calculus of variations, Springer Science & Business Media, 2009.
  63. J. Katsikadelis, M. Nerantzaki, G. Tsiatas, The analog equation method for large deflection analysis of membranes. A boundary-only solution, Computational Mechanics, 27 (2001) 513-523.
  64. H. Wang, Q.H. Qin, Y. Kang, A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media, Archive of Applied Mechanics, 74 (2005) 563-579.
  65. H. Wang, Q.H. Qin, Y. Kang, The method of fundamental solutions with radial basis functions approximation for thermoelastic analysis, 46, S1 (2006) 46-51.
  66. R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Advances in Computational Mathematics, 3 (1995) 251-264.
  67. C. Chao, M. Shen, On bonded circular inclusions in plane thermoelasticity, Journal of applied mechanics, 64 (1997) 1000-1004.
  68. V. Kupradze, M. Aleksidze, The method of functional equations for the approximate solution of certain boundary value problems, USSR Computational Mathematics and Mathematical Physics, 4 (1964) 82-126.
  69. P. Mitic, Y.F. Rashed, Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources, Engineering Analysis with Boundary Elements, 28 (2004) 143-153.
  70. H. Wang, Q.H. Qin, Y. Kang, A meshless model for transient heat conduction in functionally graded materials, Computational mechanics, 38 (2006) 51-60.
  71. H. Wang, Q.H. Qin, Some problems with the method of fundamental solution using radial basis functions, Acta Mechanica Solida Sinica, 20 (2007) 21-29.
  72. D. Young, S. Jane, C. Fan, K. Murugesan, C. Tsai, The method of fundamental solutions for 2D and 3D Stokes problems, Journal of Computational Physics, 211 (2006) 1-8.
  73. L. Marin, D. Lesnic, The method of fundamental solutions for nonlinear functionally graded materials, International journal of solids and structures, 44 (2007) 6878-6890.
  74. J. Berger, P. Martin, V. Manti?, L. Gray, Fundamental solutions for steady-state heat transfer in an exponentially graded anisotropic material, Zeitschrift für angewandte Mathematik und Physik ZAMP, 56 (2005) 293-303.
  75. L. Gray, T. Kaplan, J. Richardson, G.H. Paulino, Green's functions and boundary integral analysis for exponentially graded materials: heat conduction, Journal Of Applied Mechanics, 70 (2003) 543-549.
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International Journal of Scientific Research in Science, Engineering and Technology

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Published

2015-04-25

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Volume 1, Issue 2, March-April-2015

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Research Articles

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This work is licensed under a Creative Commons Attribution 4.0 International License.

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[1]
Yi Xiao, " Recent developments in T-Trefftz and F-Trefftz Finite Element Methods, International Journal of Scientific Research in Science, Engineering and Technology(IJSRSET), Print ISSN : 2395-1990, Online ISSN : 2394-4099, Volume 1, Issue 2, pp.441-459, March-April-2015.