SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations

Computational Mechanics

Volumn 41, Issue 4, 2008, Pages 527-545

  Batra, R C ^a   Zhang, G M ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Error norm; MLPG; Stress concentration; Strong and weak formulations; Symmetric smoothed particle hydrodynamics (SSPH) basis

Indexed keywords

BOUNDARY VALUE PROBLEMS; DIFFERENTIATION (CALCULUS); ERROR ANALYSIS; FINITE ELEMENT METHOD; FUNCTION EVALUATION; STRESS CONCENTRATION;

ERROR NORMS; MESHLESS METHODS; STRONG AND WEAK FORMULATIONS; SYMMETRIC SMOOTHED PARTICLE HYDRODYNAMICS (SSPH) BASIS;

HYDRODYNAMICS;

EID: 36949023614     PISSN: 01787675     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00466-007-0209-3     Document Type: Article

References (35)

1. Lucy LB (1977). A numerical approach to the testing of the fission hypothesis. Astron J 82: 1013-1024
2. Chen JK, Beraun JE and Jin CJ (1999). An improvement for tensile instability in smoothed particle hydrodynamics. Comput Mech 23: 279-287
3. Chen JK, Beraun JE and Jin CJ (1999). Completeness of corrective smoothed particle method for linear elastodynamics. Comput Mech 24: 273-285
4. Liu WK, Jun S and Zhang YF (1995). Reproducing kernel particle methods. Int J Num Meth Fluids 20: 1081-1106
5. Liu WK, Jun S, Li S, Adee J and Belytschko T (1995). Reproducing kernel particle methods for structural dynamics. Int J Num Meth Eng 38: 1655-1679
6. Chen JS, Pan C, Wu CT and Liu WK (1996). Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Method Appl Mech Eng 139: 195-227
7. Zhang GM and Batra RC (2004). Modified smoothed particle hydrodynamics method and its application to transient problems. Comput Mech 34: 137-146
8. Batra RC and Zhang GM (2004). Analysis of adiabatic shear bands in elasto-thermo- viscoplastic materials by modified smoothed-particle hydrodynamics (MSPH) method. J Comput Phys 201: 172-190
9. Zhang GM and Batra RC (2007). Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method. J Comput Phys 222: 374-390
10. Batra RC and Zhang GM (2007). Search algorithm and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method. Comput Mech 40: 531-546
11. Zhang GM, Batra RC (2006) Symmetric smoothed particle hydrodynamics (SSPH) method and its application to 2-D elastic problems (submitted)
12. Belytschko T, Lu YY, Gu L and Tabbara M (1995). Element-free Galerkin methods for static and dynamic fracture. Int J Solids Struct 32: 2547-2570
13. Belytschko T, Lu YY and Gu L (1994). Element-free Galerkin methods. Int J Num Meth Eng 37: 229-256
14. Atluri SN, Kim H-G and Cho JY (1999). A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods. Comput Mech 24: 348-372
15. Kim DW and Liu WK (2006). Maximum principle and convergence analysis for the meshfree point collocation method. SIAM J Numer Anal 44: 515-539
16. Kim DW, Yoon YC, Liu WK and Belytschko T (2007). Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative. J Comput Phys 221: 370-394
17. Li SF and Liu WK (1999). Reproducing kernel hierarchical partition of unity, part I-formulation and theory. Int J Num Meth Eng 45: 251-288
18. Li SF and Liu WK (1999). Reproducing kernel hierarchical partition of unity, part II-applications. Int J Num Meth Eng 45: 289-317
19. Belytschko T, Krongauz Y, Dolbow J and Gerlach C (1998). On the completeness of meshfree particle methods. Int J Num Meth Eng 43: 785-819
20. Gunther F, Liu WK, Diachin D and Christon MA (2000). Multi-scale meshfree parallel computations for viscous, compressible flows. Comput Methods Appl Mech Eng 190: 279-303
21. Ferreira AJM (2003). Thick composite beam analysis using a global meshless approximation based on radial basis functions. Mech Adv Mater Struct 10: 271-284
22. Qian LF, Batra RC and Chen LM (2003). Elastostatic deformations of a thick plate by using a higher-order shear and normal deformable plate theory and two meshless local Petrov-Galerkin (MLPG) methods. CMES Comput Model Eng Sci 4: 161-175
23. Batra RC, Porfiri M and Spinello D (2004). Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction. Int J Num Meth Eng 61: 2461-2479
24. Xiao JR and McCarthy MA (2003). A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions. Comput Mech 31: 301-315
25. Lancaster P (1969). Theory of matrices. Academic, New York
26. David PJ (1963). Interpolation and approximation. Dover Publications, New York
27. Lancaster P and Salkauskas K (1981). Surface generated by moving least squares method. Math Comput 37: 141-158
28. Timoshenko SP and Goodier JN (1970). Theory of elasticity, 3rd edn. McGraw-Hill, New York
29. Batra RC and Liang XO (1997). Finite dynamic deformations of smart structures. Comput Mech 20: 427-438
30. Qian LF, Batra RC and Chen LM (2004). Static and dynamic deformations of thick functionally graded elastic plate by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method. Composites Part B 35: 685-697
31. Qian LF and Batra RC (2004). Transient thermoelastic deformations of a thick functionally graded plate. J Therm Stresses 27: 705-740
32. Qian LF, Batra RC and Chen LM (2003). Free and forced vibrations of thick rectangular plates by using higher-order shear and normal deformable plate theory and meshless Petrov-Galerkin (MLPG) method. CMES: Comput Model Eng Sci 4: 519-534
33. Qian LF, Batra RC and Chen LM (2004). Analysis of cylindrical bending thermoelastic deformations of functionally graded plates by a meshless local Petrov-Galerkin method. Comput Mech 33: 263-273
34. Qian LF and Batra RC (2005). Design of bidirectional functionally graded plate for optimal natural frequencies. J Sound Vib 280: 415-424
35. Qian LF and Batra RC (2005). Three-dimensional transient heat conduction in a functionally graded thick plate with a higher-order plate theory and a meshless local Petrov-Galerkin method. Comput Mech 35: 214-226