Skip to main content
Log in

Shift boundary material point method: an image-to-simulation workflow for solids of complex geometries undergoing large deformation

Computational Particle Mechanics Aims and scope Submit manuscript

Abstract

We introduce a mathematical framework designed to enable a simple image-to-simulation workflow for solids of complex geometries in the geometrically nonlinear regime. While the material point method is used to circumvent the mesh distortion issues commonly exhibited in Lagrangian meshes, a shifted domain technique originated from Main and Scovazzi (J Comput Phys 372:972–995, 2018) is used to represent the boundary conditions implicitly via a level set or signed distance function. Consequently, this method completely bypasses the need to generate high-quality conformal mesh to represent complex geometries and therefore allows modelers to select the space of the interpolation function without the constraints due to the geometric need. This important simplification enables us to simulate deformation of complex geometries inferred from voxel images. Verification examples on deformable body subjected to finite rotation have shown that the new shifted domain material point method is able to generate frame-indifferent results. Meanwhile, simulations using micro-CT images of a Hostun sand have demonstrated that this method is able to reproduce the quasi-brittle damage mechanisms of single grain without the excessively concentrated nodes commonly displayed in conformal meshes that represent 3D objects with local fine details.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Abe K, Soga K, Bandara S (2013) Material point method for coupled hydromechanical problems. J Geotech Geoenviron Eng 140(3):04013033

    Google Scholar 

  2. Andersen S, Andersen L (2010) Modelling of landslides with the material-point method. Comput Geosci 14(1):137–147

    MATH  Google Scholar 

  3. Bardenhagen SG, Brackbill JU, Sulsky D (2000) The material-point method for granular materials. Comput Methods Appl Mech Eng 187(3–4):529–541

    MATH  Google Scholar 

  4. Belytschko T, Liu WK, Moran B, Elkhodary K (2013) Nonlinear Finite Elements for Continua and Structures. Wiley, London

    MATH  Google Scholar 

  5. Charlton TJ, Coombs WM, Augarde CE (2017) iGIMP: an implicit generalised interpolation material point method for large deformations. Comput Struct 190:108–125

    Google Scholar 

  6. Chen J-S, Pan C, Roque CMOL, Wang H-P (1998) A lagrangian reproducing kernel particle method for metal forming analysis. Comput Mech 22(3):289–307

    MATH  Google Scholar 

  7. Chen J-S, Hillman M, Chi S-W (2017) Meshfree methods: progress made after 20 years. J Eng Mech 143(4):04017001

    Google Scholar 

  8. Cortis M, Coombs W, Augarde C, Brown M, Brennan A, Robinson S (2018) Imposition of essential boundary conditions in the material point method. Int J Numer Methods Eng 113(1):130–152

    MathSciNet  Google Scholar 

  9. Cuitino A, Ortiz M (1992) A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics. Eng Comput 9(4):437–451

    Google Scholar 

  10. de Borst R, Crisfield MA, Remmers JJC, Verhoosel CV et al (2012) Non-linear finite element analysis of solids and structures. Wiley, London

    MATH  Google Scholar 

  11. de Souza Neto EA, Peric D, Owen DRJ (2011a) Computational methods for plasticity: theory and applications. Wiley, London

    Google Scholar 

  12. de Souza Neto EA, Peric D, Owen DRJ (2011b) Computational methods for plasticity: theory and applications. Wiley, London

    Google Scholar 

  13. Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193(12–14):1257–1275

    MathSciNet  MATH  Google Scholar 

  14. Gong M (2015) Improving the material point method. Ph.D. dissertation, University of New Mexico

  15. Guan PC, Chen JS, Wu Y, Teng H, Gaidos J, Hofstetter K, Alsaleh M (2009) Semi-lagrangian reproducing kernel formulation and application to modeling earth moving operations. Mech Mater 41(6):670–683

    Google Scholar 

  16. Guan PC, Chi SW, Chen JS, Slawson TR, Roth MJ (2011) Semi-lagrangian reproducing kernel particle method for fragment-impact problems. Int J Impact Eng 38(12):1033–1047

    Google Scholar 

  17. Guilkey JE, Weiss JA (2003) Implicit time integration for the material point method: quantitative and algorithmic comparisons with the finite element method. Int J Numer Methods Eng 57(9):1323–1338

    MATH  Google Scholar 

  18. Gupta R, Ando E, Salager S, Wang K, Sun WC (2018a) Open source database for validating and falsifying discrete mechanics models using synthetic granular materials—Part I: experimental tests with particles manufactured by a 3D printer

  19. Gupta R, Salager S, Wang K, Sun WC (2018b) Open-source support toward validating and falsifying discrete mechanics models using synthetic granular materials—Part I: experimental tests with particles manufactured by a 3D printer. Acta Geotechn, pp 1–15

  20. Jirásek M (2007) Nonlocal damage mechanics. Revue européenne de génie civil 11(7–8):993–1021

    Google Scholar 

  21. Kimmel R, Shaked D, Kiryati N, Bruckstein AM (1995) Skeletonization via distance maps and level sets. Comput Vis Image Underst 62(3):382–391

    Google Scholar 

  22. Kuhn MR, Sun WC, Wang Q (2015) Stress-induced anisotropy in granular materials: fabric, stiffness, and permeability. Acta Geotech 10(4):399–419

    Google Scholar 

  23. Kularathna S, Soga K (2017) Implicit formulation of material point method for analysis of incompressible materials. Comput Methods Appl Mech Eng 313:673–686

    MathSciNet  Google Scholar 

  24. Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55(1):1–34

    Google Scholar 

  25. Liu C, Sun Q, Yang Y (2017) Multi-scale modelling of granular pile collapse by using material point method and discrete element method. Procedia Eng 175:29–35

    Google Scholar 

  26. Liu Y, Sun WC, Fish J (2016a) Determining material parameters for critical state plasticity models based on multilevel extended digital database. J Appl Mech 83(1):011003

    Google Scholar 

  27. Liu Y, Sun WC, Yuan Z, Fish J (2016b) A nonlocal multiscale discrete-continuum model for predicting mechanical behavior of granular materials. Int J Numer Methods Eng 106(2):129–160

    MathSciNet  MATH  Google Scholar 

  28. Lopes A, Brodlie K (2003) Improving the robustness and accuracy of the marching cubes algorithm for isosurfacing. IEEE Trans Vis Comput Gr 9(1):16–29

    Google Scholar 

  29. Main A, Scovazzi G (2018a) The shifted boundary method for embedded domain computations. Part I: poisson and stokes problems. J Comput Phys 372:972–995

    MathSciNet  MATH  Google Scholar 

  30. Main A, Scovazzi G (2018b) The shifted boundary method for embedded domain computations. Part II: linear advection–diffusion and incompressible Navier–Stokes equations. J Comput Phys

  31. Mazars J (1984) Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture du béton de structure. THESE DE DOCTEUR ES SCIENCES PRESENTEE A L’UNIVERSITE PIERRE ET MARIE CURIE-PARIS 6

  32. Mota A, Sun WC, Ostien JT, Foulk JW, Long KN (2013) Lie-group interpolation and variational recovery for internal variables. Comput Mech 52(6):1281–1299

    MathSciNet  MATH  Google Scholar 

  33. Mueller R, Gross D, Maugin GA (2004) Use of material forces in adaptive finite element methods. Comput Mech 33(6):421–434

    MATH  Google Scholar 

  34. Na S, Sun W (2017) Computational thermo-hydro-mechanics for multiphase freezing and thawing porous media in the finite deformation range. Comput Methods Appl Mech Eng 318:667–700

    MathSciNet  Google Scholar 

  35. Nair A, Roy S (2012) Implicit time integration in the generalized interpolation material point method for finite deformation hyperelasticity. Mech Adv Mater Struct 19(6):465–473

    Google Scholar 

  36. Nitsche J (1971) Über ein variationsprinzip zur lösung von dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36(1):9–15. https://doi.org/10.1007/BF02995904 ISSN 1865-8784

    Article  MathSciNet  MATH  Google Scholar 

  37. Ortiz M (2002) Course notes for computational solid mechanics

  38. Papadrakakis M (1981) A method for the automatic evaluation of the dynamic relaxation parameters. Comput Methods Appl Mech Eng 25(1):35–48

    MathSciNet  MATH  Google Scholar 

  39. Qinami A, Bryant EC, Sun WC, Kaliske M (2019) Circumventing mesh bias by r- and h-adaptive techniques for variational eigen-fracture. Int J Fract

  40. Schillinger D, Ruess M (2015) The finite cell method: a review in the context of higher-order structural analysis of cad and image-based geometric models. Arch Comput Methods Eng 22(3):391–455

    MathSciNet  MATH  Google Scholar 

  41. Sirjani A, Cross GR (1991) On representation of a shape’s skeleton. Pattern Recognit Lett 12(3):149–154

    Google Scholar 

  42. Song T, Main A, Scovazzi G, Ricchiuto M (2018) The shifted boundary method for hyperbolic systems: embedded domain computations of linear waves and shallow water flows. J Comput Phys 369:45–79

    MathSciNet  MATH  Google Scholar 

  43. Sulsky D, Kaul A (2004) Implicit dynamics in the material-point method. Comput Methods Appl Mech Eng 193(12–14):1137–1170

    MathSciNet  MATH  Google Scholar 

  44. Sulsky D, Gong M (2016) Improving the material-point method. In: Innovative numerical approaches for multi-field and multi-scale problems. Springer, Berlin, 217–240

    Google Scholar 

  45. Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118(1–2):179–196

    MathSciNet  MATH  Google Scholar 

  46. Sulsky D, Zhou S-J, Schreyer HL (1995) Application of a particle-in-cell method to solid mechanics. Comput Phys Commun 87(1–2):236–252

    MATH  Google Scholar 

  47. Sulsky D, Schreyer H, Peterson K, Kwok Ron, Coon M (2007) Using the material-point method to model sea ice dynamics. J Geophys Res: Oceans 112(C2)

  48. Sun W (2015) A stabilized finite element formulation for monolithic thermo-hydro-mechanical simulations at finite strain. Int J Numer Methods Eng 103(11):798–839

    MathSciNet  MATH  Google Scholar 

  49. Sun W, Andrade JE, Rudnicki JW (2011a) Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability. Int J Numer Methods Eng 88(12):1260–1279

    MathSciNet  MATH  Google Scholar 

  50. Sun W, Andrade JE, Rudnicki JW, Eichhubl P (2011b) Connecting microstructural attributes and permeability from 3D tomographic images of in situ shear-enhanced compaction bands using multiscale computations. Geophys Res Lett 38(10)

    Google Scholar 

  51. Sun WC, Wong T (2018) Prediction of permeability and formation factor of sandstone with hybrid lattice boltzmann/finite element simulation on microtomographic images. Int J Rock Mech Min Sci 106:269–277

    Google Scholar 

  52. Sun WC, Kuhn MR, Rudnicki JW (2013a) A multiscale DEM-LBM analysis on permeability evolutions inside a dilatant shear band. Acta Geotechnica 8(5):465–480

    Google Scholar 

  53. Sun WC, Ostien JT, Salinger AG (2013b) A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain. Int J Numer Anal Methods Geomech 37(16):2755–2788

    Google Scholar 

  54. Truster TJ, Nassif O (2017) Variational projection methods for gradient crystal plasticity using lie algebras. Int J Numer Methods Eng 110(4):303–332

    MathSciNet  MATH  Google Scholar 

  55. Wang B, Vardon PJ, Hicks MA, Chen Z (2016) Development of an implicit material point method for geotechnical applications. Comput Geotech 71:159–167

    Google Scholar 

  56. Wang K, Sun W (2016) A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain. Comput Methods Appl Mech Eng 304:546–583

    MathSciNet  MATH  Google Scholar 

  57. Wang K, Sun WC (2018) A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning. Comput Methods Appl Mech Eng 334:337–380

    MathSciNet  Google Scholar 

  58. Wang K, Sun WC (2019) Meta-modeling game for deriving theory-consistent, microstructure-based traction-separation laws via deep reinforcement learning. Comput Methods Appl Mech Eng 346:216–241

    MathSciNet  Google Scholar 

  59. Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin

    MATH  Google Scholar 

  60. Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery (SPR) and adaptive finite element refinement. Comput Methods Appl Mech Eng 101(1–3):207–224

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This paper is dedicated to Professor Jiun-Shyan Chen on the occasion of his 60th birthday. This work is supported by the Earth Materials and Processes program from the US Army Research Office under grant contract W911NF-18-2-0306, the Dynamic Materials and Interactions Program from the Air Force Office of Scientific Research under grant contract FA9550-17-1-0169, the nuclear energy university program from Department of Energy under grant contract DE-NE0008534 and the Mechanics of Material program at National Science Foundation under grant contract CMMI-1462760. These supports are gratefully acknowledged. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the sponsors, including the Army Research Laboratory or the US Government. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to WaiChing Sun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

1.1 Linearization

We linearize all the terms varying with \(\varvec{u}\) in (12) to implement the Newton–Raphson iteration. We here only show the procedure of linearization of the terms containing \(\varvec{\sigma } = \varvec{\sigma ({\varvec{u}}})\). There are two main approaches to linearize the weak form of the balance of linear momentum. One is to derive the rate of internal energy using an objective stress rate, in which the tangent stiffness matrix is composed by material tangent stiffness and geometric stiffness, as shown in [4]. In [12], the measurement of stress, such as \(\varvec{P}\), is a function of the displacement through the deformation gradient:

$$\begin{aligned} \varvec{P} = \varvec{P} \left( \varvec{F} \left( \varvec{u} \right) \right) . \end{aligned}$$
(53)

The directional derivative of \(\varvec{P}\) at \(\varvec{u}^*\) therefore is:

$$\begin{aligned} D \varvec{P}_{\varvec{u}= \varvec{u}^*} \cdot \delta {\varvec{u}} = \left. \frac{\hbox {d}}{\hbox {d}\epsilon } \right| _{\epsilon =0} \frac{\partial \varvec{P}}{\partial \varvec{F}} \frac{\partial \varvec{F} (\varvec{u}^* + \epsilon \delta \varvec{u} )}{\partial \epsilon } = \varvec{A} : {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}}. \end{aligned}$$
(54)

To some extent, the tangent modulus \(\varvec{A} = \frac{\partial \varvec{P}}{\partial \varvec{F}}\) is in a general form. We can also obtain

$$\begin{aligned} D \left[ \int _{\mathcal {B}_0} \varvec{\varvec{P}}:{{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\varvec{\eta }} \hbox {d}V\right] \cdot \delta \varvec{u}= & {} \int _{\mathcal {B}_0} \varvec{A} : {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}} : {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\varvec{\eta }} \hbox {d}V, \nonumber \\\end{aligned}$$
(55)
$$\begin{aligned} D \left[ \int _{\Gamma _0} \varvec{\eta } \cdot \varvec{\varvec{P}} \cdot \varvec{N} \hbox {d}A \right] \cdot \delta \varvec{u}= & {} \int _{\Gamma _0} \varvec{\eta } \cdot \varvec{A}: {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}} \cdot \varvec{N} \hbox {d}A.\nonumber \\ \end{aligned}$$
(56)

Since the material and spatial virtual work functional are equivalent, we can transform (55) and (56) into

$$\begin{aligned} D \left[ \int _{\mathcal {B}} \varvec{\sigma } :{{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\varvec{\eta }} \hbox {d}V\right] \cdot \delta \varvec{u} =&D \left[ \int _{\mathcal {B}_0} \varvec{\varvec{P}} :{{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\varvec{\eta }} \hbox {d}V\right] \cdot \delta \varvec{u}\nonumber \\ =&\int _{\mathcal {B}} \frac{1}{J} \varvec{A} : ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} \varvec{F}): ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\varvec{\eta }} \varvec{F}) \hbox {d}V \nonumber \\ =&\int _{\mathcal {B}} \varvec{a} : ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} ) : ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\varvec{\eta }}) \hbox {d}V, \end{aligned}$$
(57)
$$\begin{aligned} D \left[ \int _{\Gamma } \varvec{\eta } \cdot \varvec{\sigma } \cdot \varvec{n} \hbox {d}A \right] \cdot \delta \varvec{u} =&D \left[ \int _{\Gamma _0} \varvec{\eta } \cdot \varvec{\varvec{P}} \cdot \varvec{N} \hbox {d}A \right] \cdot \delta \varvec{u}\nonumber \\ =&\int _{\Gamma _0} \varvec{\eta } \cdot \varvec{A}: {{\,\mathrm{\nabla ^{\varvec{X}}}\,}}{\delta \varvec{u}} \cdot \varvec{N} \hbox {d}A \nonumber \\ =&\int _{\Gamma } \frac{1}{J} \varvec{\eta } \cdot \varvec{A}: ({{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} \varvec{F}) \cdot \varvec{F}^T \cdot \varvec{n} \hbox {d}A \nonumber \\ =&\int _{\Gamma } \varvec{\eta } \cdot \varvec{a}: {{\,\mathrm{\nabla ^{\varvec{x}}}\,}}{\delta \varvec{u}} \cdot \varvec{n} \hbox {d}A, \end{aligned}$$
(58)

where \(\varvec{a}\) is defined in (35).

Fig. 18
figure 18

Model of a bar under a uniaxial loading

Fig. 19
figure 19

Stress–strain curves with different meshes for local (loc) and nonlocal damage (non) models

1.2 Verification of the local nonlocal damage model

We here adopt a uniaxial loading example to verify the nonlocal damage model. As shown in Fig. 18, a bar with a length of 10 m is fixed on the left bound and subjected to an incremental displacement on the right bound. The material is described by an isotropic elastic damage model. The elastic modulus is 10 GPa, Poisson’s ratio is 0.2, and \(\varepsilon _0\) and \(\varepsilon _f\) in (30) are \(1\times 10^{-4}\) and \(2\times 10^{-3}\), respectively. We set the size of element as \(h=0.5, 1\) and 2m. The internal length for the integral regularization is 2.2 m. Figure 19 plots the curves of \(\varepsilon -\sigma \) for different mesh sizes and constitutive models. We can see that the results are mesh-dependent for local damage model and less dissipation energy for the finer grid. On the contrary, the results are identical for different meshes with the nonlocal damage model. We further compare the distribution of strain along the horizontal axis and find that the distribution is almost uniform for the nonlocal damage model and the strain localizes at the rightmost element for the local damage model. We thus believe our implementation of the nonlocal damage model is correct.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, C., Sun, W. Shift boundary material point method: an image-to-simulation workflow for solids of complex geometries undergoing large deformation. Comp. Part. Mech. 7, 291–308 (2020). https://doi.org/10.1007/s40571-019-00239-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40571-019-00239-y

Keywords

Navigation