Computational Mechanics

, Volume 63, Issue 5, pp 1069–1082

# A stochastic material point method for probabilistic dynamics and reliability

• Weidong Chen
• Yaqin Shi
• Han Yan
• Jingxin Ma
• Yuzhuo Yang
• Chunlong Xu
Original Paper

## Abstract

A stochastic material point method is proposed for reliability analysis of nonlinear structure subjected to explosions involving spatially varying random material properties. A random field representing material properties is discretized into a set of random variables with statistical properties of the random field. According to the failure criterion of nonlinear structure, the limit state function of a material point is established. The first-order reliability method is employed to predict the full probabilistic characteristics of material points. Besides, taking ship protection structure as an example, the failure mode of nonlinear structure is established and the model is implemented into the reliability analysis of ship protection structure subjected to underwater explosions. Numerical examples are presented to examine the accuracy and convergence of the stochastic material point method. Monte Carlo simulation is used as a validation tool, and good agreement is obtained between the results of the proposed method and Monte Carlo simulation.

## Keywords

Stochastic material point method Limit state function Failure mode Nonlinear structure Structural reliability

## References

1. 1.
Shinozuka M, Deodatis G (1988) Response variability of stochastic finite element systems. J Eng Mech 114(3):499–519
2. 2.
Graham LL, Siragy EF (2001) Stochastic finite-element analysis for elastic buckling of stiffened panels. J Eng Mech 127(1):91–97
3. 3.
Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198:1031–1051
4. 4.
Tao L, Song H, Chakrabarti S (2007) Scaled boundary FEM solution of short-crested wave diffraction by a vertical cylinder. Comput Methods Appl Mech Eng 197:232–242
5. 5.
Natarajan S, Wang JC, Song CM, Birk C (2015) Isogeometric analysis enhanced by the scaled boundary finite element method. Comput Methods Appl Mech Eng 283:733–762
6. 6.
Song CM, Wolf JP (2002) Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method. Comput Struct 80:183–197
7. 7.
Long XY, Jiang C, Yang C, Han X, Gao W, Liu J (2016) A stochastic scaled boundary finite element method. Comput Methods Appl Mech Eng 308:23–46
8. 8.
Long XY, Jiang C, Yang C, Han X, Gao W (2015) Stochastic response analysis of the scaled boundary finite element method and application to probabilistic fracture mechanics. Comput Struct 153:185–200
9. 9.
Rabczuk T, Belytschko T (2017) A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Methods Appl Mech Eng 318:762–782
10. 10.
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 199:2437–2455
11. 11.
Arun C, Rao B, Kumar MS (2007) An application of stochastic meshfree method in the field of fracture mechanics. In: Proceedings of international symposium on computational mechanics. Springer, p 227Google Scholar
12. 12.
Sellountos Euripides J, Sequeira Adélia (2008) An advanced meshless LBIE/RBF method for solving two-dimensional incompressible fluid flows. Comput Mech 41(5):617–631
13. 13.
Lucy LB (1977) A numerical approach to testing the fission hypothesis. Astron J 82:1013–1024
14. 14.
Monaghan JJ (1988) An introduction to SPH. Comput Phys Commun 48:89–96
15. 15.
Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318
16. 16.
Lu YY, Belytschko T, Gu L (1994) A new implementation of the element free Galerkin method. Comput Methods Appl Mech Eng 113:397–414
17. 17.
Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256
18. 18.
Melenk JM, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:280–314
19. 19.
Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106
20. 20.
Liu WK, Li S, Belytschko T (1997) Moving least square kernel Galerkin method-part I: methodology and convergence. Comput Methods Appl Mech Eng 143:422–433
21. 21.
Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–186
22. 22.
Sulsky D, Schreyer HL (1996) Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems. Comput Methods Appl Mech Eng 139:409–429
23. 23.
Sulsky D, Zhou SJ, Schreyer HL (1995) Application of a particle-in-cell method to solid mechanics. Comput Phys Commun 87(1):236–252
24. 24.
Andersen S, Andersen L (2010) Analysis of spatial interpolation in the material-point method. Comput Struct 88(7–8):506–518
25. 25.
Ching HK, Batra RC (2001) Determination of crack tip fields in linear elastostatics by the meshless local Petrov-Galerkin (MLPG) method. Comput Model Eng Sci 2:273–289Google Scholar
26. 26.
Mason M, Chen K, Hu PG (2014) Material point method of modelling and simulation of reacting flow of oxygen. Int J Comput Fluid Dyn 28:420–427
27. 27.
Ma J, Wang D, Randolph MF (2014) A new contact algorithm in the material point method for geotechnical simulations. Int J Numer Anal Methods Geomech 38(11):1197–1210
28. 28.
Nairn JA, Guilkey JE (2015) Axisymmetric form of the generalized interpolation material point method. Int J Numer Methods Eng 101(2):127–147
29. 29.
Ma S, Zhang X, Qiu XM (2009) Comparison study of MPM and SPH in modeling hypervelocity impact problems. Int J Impact Eng 36(2):272–282
30. 30.
Tao J, Zhang HG, Zheng YZ, Chen Z (2018) Development of generalized interpolation material point method for simulating fully coupled thermomechanical failure evolution. Comput Methods Appl Mech Eng 332:325–342
31. 31.
Hu W, Chen Z (2003) A multi-mesh MPM for simulating the meshing process of spur gears. Comput Struct 81(20):1991–2002
32. 32.
Gan Y, Chen Z, Montgomery-Smith S (2011) Improve material point method for simulating the zona failure response in piezo-assisted intracytoplasmic sperm injection. Comput Model Eng Sci 73(1):45–75
33. 33.
Lu MK, Zhang JY, Zhang HW, Zheng YG, Chen Z (2018) Time-discontinuous material point method for transient problems. Comput Methods Appl Mech Eng 328:663–685
34. 34.
Jiang S, Chen Z, Sewell TD, Gan Y (2015) Multiscale simulation of the responses of discrete nanostructures to extreme loading conditions based on the material point method. Comput Methods Appl Mech Eng 297:219–238
35. 35.
Von Neumann J, Richtmyer RD (1950) A method for the numerical calculation of hydrodynamical shocks. J Appl Phys 21(3):232–257
36. 36.
Landshoff R (1955) A numerical method for treating fluid flow in the presence of shocks. Los Alamos Scientific Laboratory, Rept. LA-1930, 1955Google Scholar
37. 37.
Ghanem RG, Spanos PD (1991) Stochastic finite elements. A spectral approach. Springer, Berlin
38. 38.
Steven Greene M, Liu Y, Chen W, Liu WK (2011) Computational uncertainty analysis in multiresolution materials via stochastic constitutive theory. Comput Methods Appl Mech Eng 200(1–4):309–325
39. 39.
Der Kiureghian A, Liu P-L (1986) Structural reliability under incomplete probability information. J Eng Mech 112(1):85–104
40. 40.
Vanmarcke EH, Grigoriu M (1983) Stochastic finite element analysis of simple beams. J Eng Mech 109(5):1203–1214
41. 41.
Liu WK, Belytschko T, Mani A (1986) Random fields finite element. Int J Numer Methods Eng 23:1831–1845
42. 42.
Deodatis G (1991) Weighted integral method I: stochastic stiffness matrix. J Eng Mech 117(8):1851–1864
43. 43.
Spanos PD, Ghanem RG (1989) Stochastic finite element expansion for random media. J Eng Mech 115(5):1035–1053
44. 44.
Johnson GR, Cook W (1985) Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 21(1):31–48
45. 45.
Bahri A, Guermazi N, Elleuch K, Ürgen M (2016) On the erosive wear of 304 L stainless steel caused by olive seed particles impact: modeling and experiments. Tribol Int 102:608–619
46. 46.
Wang XM, Shi J (2013) Validation of Johnson–Cook plasticity and damage model using impact experiment. Int J Impact Eng 60:67–75
47. 47.
Banerjee A, Dhar S (2015) Determination of Validation of Johnson–Cook plasticity and damage model using impact experiment. constants and numerical modelling of Charpy impact test of armour steel. Mater Sci Eng A 640:200–209
48. 48.
Tenorio M, Pelegri AA (2013) Interfacial debonding of glass single fiber composites using the Johnson–Cook failure model. In: ASME. ASME international mechanical engineering congress and exposition, volume 15: safety, reliability and risk; Virtual Podium (Posters): V015T16A023.
49. 49.
Shams A, Mashayekhi M (2012) Improvement of orthogonal cutting simulation with a nonlocal damage model. Int J Mech Sci 61:88–96
50. 50.
Ragnar L, Senad R, Lennart Josefson B (2016) Mesh objective continuum damage models for ductile fracture. Int J Numer Methods Eng 106(10):840–860
51. 51.
Hancock JW, Mackenzie AC (1976) On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J Mech Phys Solids 24:147–169
52. 52.
Imai K, Frangopol DM (2000) Geometrically nonlinear finite element reliability analysis of structural systems. I: theory. Comput Struct 77:677–691
53. 53.
Lopez RH, Beck AT (2012) Reliability-based design optimization strategies based on FORM: a review. J Braz Soc Mech Sci Eng 34(4):506–514
54. 54.
Koduru SD, Haukaas T (2010) Feasibility of FORM in finite element reliability analysis. Struct Saf 32(2):145–153
55. 55.
Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidiscip Optim 41(2):277–294
56. 56.
Wang L, Wang XJ, Xia Y (2014) Hybrid reliability analysis of structures with multi-source uncertainties. Acta Mech 225(2):413–430
57. 57.
Jiang C, Han S, Ji M, Han X (2015) A new method to solve the structural reliability index based on homotopy analysis. Acta Mech 226:1067–1083
58. 58.
Liu Y, Meng LL, Liu K, Zhang YM (2016) Chatter reliability of milling system based on first-order second-moment method. Int J Adv Manuf Technol 87(1–4):801–809
59. 59.
Liu N, Tang WH (2004) System reliability evaluation of nonlinear continuum structures—a probabilistic FEM approach. Finite Elem Anal Des 40(5–6):595–610
60. 60.
61. 61.
Zhang J, Shi XH (2017) Experimental study on the response of multi-layered protective structure subjected to underwater contact explosions. Int J Impact Eng 100:23–34
62. 62.
Charki A, Bigaud D, Guérin F (2013) Behavior analysis of machines and system air hemispherical spindles using finite element modeling. Ind Lubr Tribol 65(4):272–283

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

• Weidong Chen
• 1
• Yaqin Shi
• 1
Email author
• Han Yan
• 1
• Jingxin Ma
• 1
• Yuzhuo Yang
• 2
• Chunlong Xu
• 1
1. 1.College of Aerospace and Civil EngineeringHarbin Engineering UniversityHarbinChina
2. 2.Architecture and Civil Engineering DepartmentCity University of Hong KongKowloonHong Kong