Computational Mechanics

, Volume 53, Issue 6, pp 1087–1103

# Extended particle difference method for weak and strong discontinuity problems: part I. Derivation of the extended particle derivative approximation for the representation of weak and strong discontinuities

Original Paper

## Abstract

In this paper, the extended particle derivative approximation (EPDA) scheme is developed to solve weak and strong discontinuity problems. In this approximation scheme, the Taylor polynomial is extended with enrichment functions, i.e. the step function, the wedge function, and the scissors function, based on the moving least squares procedure in terms of nodal discretization. Throughout numerical examples, we demonstrate that the EPDA scheme reproduces weak and strong discontinuities in a singular solution quite well, and effectively copes with the difficulties in computing the derivatives of the singular solution. The governing partial differential equations, including the interface conditions, are directly discretized in terms of the EPDA scheme, and the total system of equations is derived from the formulation of difference equations which is constructed at the nodes and points representing the problem domain and the interfacial boundary, respectively.

## Keywords

Extended particle difference method (EPDM) Extended particle derivative approximation (EPDA) Interfacial singularity Strong formulation Second-order accuracy

## References

1. 1.
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150
2. 2.
Ventura G, Xu J, Belytschko T (2002) A vector level set method and new discontinuity approximations for crack growth by EFG. Int J Numer Methods Eng 54:923–944
3. 3.
Zi G, Belytschko T (2003) New crack-tip elements for XFEM and applications to cohesive cracks. Int J Numer Methods Eng 57:2221–2240
4. 4.
Tu C, Peskin CS (1992) Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods. SIAM J Sci Stat Comput 13:1361–1376
5. 5.
LeVeque RJ, Li Z (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31:1019–1044
6. 6.
Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318
7. 7.
Krongauz Y, Belytschko T (1997) Consistent pseudo-derivatives in meshless methods. Comput Methods Appl Mech Eng 146:371–386
8. 8.
Krongauz Y, Belytschko T (1997) A Petrov–Galerkin diffuse element method (PG DEM) and its comparison to EFG. Comput Mech 19:327–333
9. 9.
Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256
10. 10.
Liu WK, Jun S, Zhang Y (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106
11. 11.
Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 30:543–574
12. 12.
Kim DW, Kim Y-S (2003) Point collocation methods using the fast moving least square reproducing kernel approximation. Int J Numer Methods Eng 56:1445–1464
13. 13.
Lee S-H, Yoon Y-C (2004) Meshfree point collocation method for elasticity and crack problem. Int J Numer Methods Eng 61:22–48
14. 14.
Oñate E, Idelsohn S, Zienkiewicz OC, Taylor RL, Sacco C (1996) A stabilized finite point method of analysis of fluid mechanics problems. Comput Methods Appl Mech Eng 139:315–346
15. 15.
Oñate E, Perazzo F, Miquel J (2001) A finite point method for elasticity problems. Comput Struct 79:2151–2163
16. 16.
Aluru NR (2000) A point collocation method based on reproducing kernel approximations. Int J Numer Methods Eng 47:1083–1121
17. 17.
Huerta A, Vidal Y, Villon P (2004) Pseudo-divergence-free element free Galerkin method for incompressible fluid flow. Comput Methods Appl Mech Eng 193:1119–1136
18. 18.
Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55:1–34
19. 19.
Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity: part I. Formulation and theory. Int J Numer Methods Eng 45:251–288
20. 20.
Fleming M, Chu YA, Moran B, Belytschko T (1997) Enrichment element-free Galerkin methods for crack tip fields. Int J Numer Methods Eng 40:1483–1504
21. 21.
Belytschko T, Fleming M (1999) Smoothing, enrichment and contact in the element-free Galerkin method. Comput Struct 71:173–195
22. 22.
Krongauz Y, Belytschko T (1998) EFG approximation with discontinuous derivatives. Int J Numer Methods Eng 41:1215–1233
23. 23.
Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61(13):2316–2343
24. 24.
Rabczuk T, Zi G (2007) A meshfree method based on the local partition of unity for cohesive cracks. Comput Mech 39(6):743–760
25. 25.
Rabczuk T, Belytschko T (2007) A three dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Methods Appl Mech Eng 196(29–30):2777–2799
26. 26.
Zi G, Rabczuk T, Wall WA (2007) Extended meshfree methods without branch enrichment for cohesive cracks. Comput Mech 40(2):367–382
27. 27.
Rabczuk T, Bordas S, Zi G (2007) A three-dimensional meshfree method for continuous multiplecrack initiation, nucleation and propagation in statics and dynamics. Comput Mech 40(3):473–495
28. 28.
Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H (2010) A simple and robust three dimensional cracking-particle method without enrichment. Comput Methods Appl Mech Eng 199(37–40):2437–2455
29. 29.
Cordes LW, Moran B (1996) Treatment of material discontinuity in the element-free Galerkin method. Comput Methods Appl Mech Eng 139:75–89
30. 30.
Luo Y, Häussler-Combe U (2002) A generalized finite-difference method based on minimizing global residual. Comput Methods Appl Mech Eng 191:1421–1438
31. 31.
Kim DW, Yoon Y-C, Liu WK, Belytschko T (2007) Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative. J Comput Phys 221:370–394
32. 32.
Kim DW, Liu WK, Yoon Y-C, Belytschko T, Lee S-H (2007) Meshfree point collocation method with intrinsic enrichment for interface problems. Comput Mech 40(6):1037–1052
33. 33.
Kim DW, Kim H-K (2004) Point collocation method based on the FMLSRK approximation for electromagnetic field analysis. IEEE Trans Magn 40(2):1029–1032
34. 34.
Legay A, Wang H-W, Belytschko T (2005) Strong and weak arbitrary discontinuities in spectral finite elements. Int J Numer Methods Eng 64:991–1008 Google Scholar
35. 35.
Chessa J, Smolinski P, Belytschko T (2002) The extended finite element method (XFEM) for solidication problems. Int J Numer Methods Eng 53(8):1959–1977
36. 36.
Benzley SE (1974) Representation of singularities with isotropic finite elements. Int J Numer Methods Eng 8:537–545
37. 37.
Gifford LN Jr, Hilton PD (1978) Stress intensity factors by enriched finite elements. Eng Fract Mech 10:485–496
38. 38.
Yoon Y-C, Lee S-H, Belytschko T (2006) Enriched meshfree collocation method with diffuse derivatives for elastic fracture. Comput Math Appl 51(8):1349–1366
39. 39.
Yoon Y-C, Kim DW (2010) Extended meshfree point collocation method for electromagnetic problems with layered singularity. IEEE Trans Magn 46(8):2951–2954
40. 40.
Swenson DV, Ingraffea A (1988) Modeling mixed-mode dynamic crack propagation using finite elements: theory and applications. Comput Mech 3:381–397
41. 41.
Martha L, Wawrzynek P, Ingraffea A (1993) Arbitrary crack propagation using solid modeling. Eng Comput 9:63–82 