Boundary particle method with high-order Trefftz Functions

Computers, Materials and Continua

Volumn 13, Issue 3, 2010, Pages 201-217

  Chen, Wen ^a   Fu, Zhuo Jia ^a,b   Qin, Qing Hua ^b  

^a HOHAI UNIVERSITY   (China)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Boundary particle method; High order Trefftz functions; Inhomogeneous Problems; Meshfree

Indexed keywords

BOUNDARY PARTICLE METHODS; HIGH-ORDER; INHOMOGENEOUS PROBLEMS; MESHFREE; TREFFTZ FUNCTIONS;

MATHEMATICAL OPERATORS;

PROBLEM SOLVING;

EID: 77951453013     PISSN: 15462218     EISSN: 15462226     Source Type: Journal    
DOI: None     Document Type: Article

References (48)

1. Chen, C. S.; Golberg, M. A.; Hon, Y. C. (1998): The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations. International Journal for Numerical Methods in Engineering, vol.43, pp. 1421-1435.
2. Chen, K. H.; Kao, J. H.; Chen, J. T. (2009): Regularized meshless method for antiplane piezoelectricity problems with multiple inclusions. CMC: Computers, Materials & Continua, vol.9, pp. 253-279.
3. Chen J. T.; Wu C. S.; Lee Y. T.; Chen K. H. (2007): On three quivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations. Compu. Math. Applic., vol.53, pp. 851-879.
4. Chen, W. (2001): Boundary knot method for Laplace and biharmonic problems. In: Proceedings of the 14th Nordic Workshop on Computational Mechanics, Sweden.
5. Chen,W.; Tanaka, M. (2002): A meshless, exponential convergence, integrationfree, and boundary-only RBF technique. Comput Math Appl, vol.43, pp. 379-391.
6. Chen, W. (2009): Singular boundary method: A novel, simple, mesh-free, boundary collocation numerical method. Acta Mechanica Solida Sinica (in Chinese), vol.30, pp. 592-599.
7. Chen, W.; Fu, Z. J. (2009): Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations. Journal of Marine Science and Technology, vol.17, pp. 157-163.
8. Chen, W.; Fu, Z. J.; Jin, B. T. (2010): A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique. Engineering Analysis with Boundary Elements, vol.34, pp. 196-205.
9. Chen, W.; Fu, Z. J.; Wei, X. (2009): Potential Problems by Singular Boundary Method Satisfying Moment Condition. CMES: Computer Modeling in Engineering & Sciences, vol.54, pp. 65-86.
10. Cheung, Y. K.; Jin,W. G.; Zienkiewicz, O. C. (1989): Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions. Commun Appl Numer Methods, vol.5, pp. 159-169.
11. Cozzano, B.S.; Rodríguez, B.S. (2007): The Trefftz Boundary Method in Viscoelasticity. CMES: Computer Modeling in Engineering & Sciences, vol.20, pp.21-33.
12. Fairweather, G.; Karageorghis, A. (1998): The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math, vol.9, pp. 69-95. (Pubitemid 128403959)
13. Fu, Z. J.; Chen, W. (2009): A truly boundary-only meshfree method applied to Kirchhoff plate bending problem. Advances in Applied Mathematics and Mechanics, vol.1, pp. 341-352.
14. Fu, Z. J.; Chen, W.; Yang W. (2009): Winkler plate bending problems by a truly boundary-only boundary particle method. Computational Mechanics, vol.44, pp. 757-763.
15. Gu, M. H.; Young, D. L.; Fan, C. M. (2009): The Method of Fundamental Solutions for One-Dimensional Wave Equations. CMC: Computers, Materials & Continua, vol.11, pp. 185-208.
16. Herrera, I. (1980): Theory of connectivity: a systematic formulation of boundary element methods. In: Brebbia CA, editor. New developments in boundary element methods (Proc 2nd Int Seminar on Recent Advances in BEM, Southampton, England, 1980). Pentech Press, pp. 45-58.
17. Hon, Y. C.;Wu, Z. M. (2000): A numerical computation for inverse determination problem. Engineering Analysis with Boundary Elements, vol.24, pp. 599-606.
18. Hu, S.P.; Young, D.L.; Fan, C.M. (2008): FDMFS for Diffusion Equation with Unsteady Forcing Function. CMES: Computer Modeling in Engineering & Sciences, vol.24, pp.1-20.
19. Jin,W. G.; Cheung, Y. K.; Zienckiewicz, O. C. (1993): Trefftz method for Kirchhoff plate bending problems. International Journal for Numerical Methods in Engineering, vol.36, pp. 765-781.
20. Jin, W. G.; Cheung, Y. K. (1999): Trefftz method applied to a moderately thick plate. International Journal for Numerical Methods in Engineering, vol.44, pp. 1011-1024.
21. Jirousek, J.; Qin, Q.H. (1996): Application of hybrid-Trefftz element approach to transient heat conduction analysis. Computers & Structures, vol.58, pp. 195-201.
22. Kamiya, N.; Kita, E. (1995): Trefftz method 70 Years. Adv Eng Softw vol. 24(1-3).
23. Kansa, E. J. (1990): Multiquadrics: A scattered data approximation scheme with applications to computational fluid-dynamics. Comput. Math. Appl., vol.19, pp. 147-161.
24. Kita, E.; Ikeda, Y.; Kamiya, N. (2005): Trefftz solution for boundary value problem of three-dimensional Poisson equation. Engineering Analysis with Boundary Elements, vol.29, pp. 383-390. (Pubitemid 40612723)
25. Lee,W. M.; Chen, J. T. (2009): Free Vibration Analysis of a Circular Plate with Multiple Circular Holes by Using the Multipole Trefftz Method. CMES: Computer Modeling in Engineering & Sciences, vol.50, pp.141-159.
26. Li, Z. C.; Lu, T. T.; Huang, H. T.; Cheng, A. H. D. (2006): Trefftz, collocation, and other boundary methods-a comparison. Numer Methods Partial Differential Equations, vol.23, pp. 93-144.
27. Li, Z. C.; Lu, T. T.; Huang, H. T.; Cheng, A. H.-D. (2009): Error Analysis of Trefftz Methods for Laplace's Equations and Its Applications. CMES: Computer Modeling in Engineering & Sciences, vol.52, pp.39-81.
28. Liu, C. S. (2007a): A meshless regularized integral equation method for Laplace equation in arbitrary interior or exterior plane domains. CMES: Computer Modeling in Engineering & Sciences, vol.19, pp. 99-109.
29. Liu, C. S. (2007b): A MRIEM for solving the Laplace equation in the doublyconnected domain. CMES: Computer Modeling in Engineering & Sciences, vol.19, pp. 145-161.
30. Liu, C. S. (2007c): A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain. CMES: Computer Modeling in Engineering & Sciences, vol.20, pp. 111-122.
31. Liu, C. S. (2008a): Improving the ill-conditioning of the method of fundamental solutions for 2d Laplace equation. CMES: Computer Modeling in Engineering & Sciences, vol.28, pp. 77-93.
32. Liu, C. S. (2008b): A Highly Accurate MCTM for Inverse Cauchy Problems of Laplace Equation in Arbitrary Plane Domains. CMES: Computer Modeling in Engineering & Sciences, vol.35, pp. 91-112.
33. Liu, C. S.; Yeih, W.; Atluri, S. N. (2009): On solving the ill-conditioned system Ax = b: general-purpose conditioners obtained from the boundary-collocation solution of the Laplace equation, using Trefftz expansions with multiple length scales. CMES: Computer Modeling in Engineering & Sciences, vol.44, pp. 281-311.
34. Marin, L. (2008): Stable MFS solution to singular direct and inverse problems associated with the Laplace equation subjected to noisy data. CMES: Computer Modeling in Engineering & Sciences, vol.37, pp. 203-242.
35. Marin, L.; Karageorghis, A. (2009): Regularized MFS-Based Boundary Identification in Two-Dimensional Helmholtz-Type Equations. CMC: Computers, Materials & Continua, vol.10, pp. 259-294.
36. Marin, L. (2009): An Alternating Iterative MFS Algorithm for the Cauchy Problem in Two-Dimensional Anisotropic Heat Conduction. CMC: Computers, Materials & Continua, vol.12, pp. 71-100.
37. Nowak, A. J.; Neves, A. C. (1994): The Multiple Reciprocity Boundary Element Method. Southampton, Computational Mechanics Publication.
38. Qin, Q. H. (2000): The Trefftz Finite and Boundary Element Method. WIT Press, Southampton.
39. Qin, Q. H. (2005): Trefftz finite element method and its applications. Applied Mechanics Reviews, vol.58, pp. 316-337.
40. Šarler, B. 2008 Chapter 15: Modified method of fundamental solutions for potential flow problems, in: C.S. Chen, A. Karageorghis, Y.S. Smyrlis (Eds.)The Method of Fundamental Solutions-A Meshless Method, Dynamic Publisher 299-321.
41. Song, R. C.; Chen, W. (2009): An investigation on the regularized meshless method for irregular domain problems. CMES: Computer Modeling in Engineering & Sciences, vol.42, pp. 59-70..
42. Trefftz, E. (1926): Ein Gegenstück zum Ritzschen Verfahren. Proc 2nd Int Cong Appl Mech, Zurich, pp. 131-137.
43. Wang, F. Z.; Ling, L.; Chen,W. (2009): Effective Condition Number for Boundary Knot Method. CMC: Computers, Materials & Continua, vol.12, pp. 57-70.
44. Wei, T.; Hon, Y. C.; Ling, L. (2007): Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Engineering Analysis with Boundary Elements, vol.31, pp. 163-175.
45. Young, D. L.; Chen, K. H.; Lee, C.W. (2005): Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys, vol.209, pp.290-321.
46. Young, D. L.; Chen, K. H.; Chen, J. T.; Kao, J. H. (2007): A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains. CMES: Computer Modeling in Engineering & Sciences, vol.19, pp. 197-221.
47. Young, D. L.; Chen, K. H.; Liu, T. Y.; Shen, L. H.;Wu, C. S. (2009): Hypersingular meshless method for solving 3D potential problems with arbitrary domain. CMES: Computer Modeling in Engineering & Sciences, vol.40, pp. 225-270.
48. Zielinski, A. P.; Zienkiewicz, O. C. (1985): Generalized finite element analysis with T-complete boundary solution function. International Journal for Numerical Methods in Engineering, vol.21, pp. 509-528.