Skip to main content
Log in

Computational aspects of dispersive computational continua for elastic heterogeneous media

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

The present manuscript focusses on computational aspects of dispersive computational continua (\(C^2\)) formulation previously introduced by the authors. The dispersive \(C^2\) formulation is a multiscale approach that showed strikingly accurate dispersion curves. However, the seemingly theoretical advantage may be inconsequential due to tremendous computational cost involved. Unlike classical dispersive methods pioneered more than a half a century ago where the unit cell is quasi-static and provides effective mechanical and dispersive properties to the coarse-scale problem, the dispersive \(C^2\) gives rise to transient problems at all scales and for all microphases involved. An efficient block time-integration scheme is proposed that takes advantage of the fact that the transient unit cell problems are not coupled to each other, but rather to a single coarse-scale finite element they are positioned in. We show that the computational cost of the method is comparable to the classical dispersive methods for short load durations.

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

Similar content being viewed by others

Notes

  1. The unit cells geometrical configuration is usually selected as a box-like, such as rectangle in 2D or cuboid in 3D. This imposes a restriction on modeling the physical domain of the problem by spatial periodicity of the unit cell if the problem is solved by direct numerical simulation (DNS). However, in the \(C^2\) approach the unit cell is a computational entity in the same way as the local (classical) Gauss points. Thus, in \(C^2\) the coarse-scale physical domain can be approximated by structured or unstructured meshes and still the computational unit cells being a box-like geometrical entity. Moreover, in \(C^2\) the computational unit cell is finite, but no larger than a coarse-scale element domain. For a more detailed discussion on determining the nonlocal quadrature points please refer to [16, 22]

  2. The unit cells do not necessarily tile the macro-scale domain. Only in special cases of a regular coarse-scale mesh and specific ratios of unit cell size to coarse-scale element the computational unit cells exactly tile the macrostructure. Their behavior is upscaled to the macroscale through the nonlocal quadrature scheme

References

  1. Abdulle A, Weinan E, Engquist B, Vanden-Eijnden E (2012) The heterogeneous multiscale method. Acta Numer 21:1–87. doi:10.1017/S0962492912000025

    Article  MATH  MathSciNet  Google Scholar 

  2. Abdulle A, Grote M, Stohrer C (2014) Finite element heterogeneous multiscale method for the wave equation: long-time effects. Multiscale Model Simul 12(3):1230–1257. doi:10.1137/13094195X

    Article  MathSciNet  Google Scholar 

  3. Achenbach JD, Herrmann G (1968) Wave motion in solids with lamellar structuring. In: ASME-dynamics of structured solids, pp 23–46

  4. Andrianov I, Bolshakov V, Danishevs’kyy V, Weichert D (2008) Higher order asymptotic homogenization and wave propagation in periodic composite materials. Proc R Soc A 464(2093):1181–1201. doi:10.1098/rspa.2007.0267

    Article  MATH  MathSciNet  Google Scholar 

  5. Auriault JL, Boutin C (2012) Long wavelength inner-resonance cut-off frequencies in elastic composite materials. Int J Solids Struct 49(23–24):3269–3281. doi:10.1016/j.ijsolstr.2012.07.002

    Article  Google Scholar 

  6. Auriault JL, Boutin C, Geindreau C (2010) Homogenization of coupled phenomena in heterogenous media. doi:10.1002/9780470612033

  7. Bedford A, Stern M (1970) On wave propagation in fiber- reinforced viscoelastic materials. J Appl Mech 37:1190–1192

    Article  Google Scholar 

  8. Chen W, Fish J (2006a) A generalized space-time mathematical homogenization theory for bridging atomistic and continuum scales. Int J Numer Methods Eng 67(2):253–271. doi:10.1002/nme.1630

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen W, Fish J (2006b) A mathematical homogenization perspective of virial stress. Int J Numer Methods Eng 67(2):189–207. doi:10.1002/nme.1622

    Article  MATH  MathSciNet  Google Scholar 

  10. Clausius R (1870) On a mechanical theorem applicable to heat. Philos Mag 122–127. doi:10.1080/14786447008640370

  11. Craster R, Kaplunov J, Pichugin A (2010a) High-frequency homogenization for periodic media. Proc R Soc A 466(2120):2341–2362. doi:10.1098/rspa.2009.0612

    Article  MATH  MathSciNet  Google Scholar 

  12. Craster R, Kaplunov J, Postnova J (2010b) High-frequency asymptotics, homogenisation and localisation for lattices. Q J Mech Appl Math 63(4):497–519. doi:10.1093/qjmam/hbq015

    Article  MATH  MathSciNet  Google Scholar 

  13. Ding Y, Liu Z, Qiu C, Shi J (2007) Metamaterial with simultaneously negative bulk modulus and mass density. Phys Rev Lett 99(9). doi:10.1103/PhysRevLett.99.093904

  14. Ye Efendiev, Jc Galvis, Hou T (2013) Generalized multiscale finite element methods (gmsfem). J Comput Phys 251:116–135. doi:10.1016/j.jcp.2013.04.045

    Article  MathSciNet  Google Scholar 

  15. Filonova V, Fafalis D, Fish J (2015) Dispersive computational continua. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2015.08.008

  16. Fish J (2013) Practical multiscaling. Wiley

  17. Fish J, Chen W (2001) Higher-order homogenization of initial/boundary-value problem. J Eng Mech 127(12):1223–1230. doi:10.1061/(ASCE)0733-9399(2001)127:12(1223)

    Article  Google Scholar 

  18. Fish J, Kuznetsov S (2010) Computational continua. Int J Numer Methods Eng 84(7):774–802. doi:10.1002/nme.2918

    Article  MATH  MathSciNet  Google Scholar 

  19. Fish J, Chen W, Li R (2007) Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions. Comput Methods Appl Mech Eng 196(4–6):908–922. doi:10.1016/j.cma.2006.08.001

    Article  MATH  MathSciNet  Google Scholar 

  20. Fish J, Li A, Yavari F (2010) Adaptive generalized mathematical homogenization framework for nanostructured materials. Int J Numer Methods Eng 83(8–9):1133–1154. doi:10.1002/nme.2895

    Article  MATH  Google Scholar 

  21. Fish J, Filonova V, Kuznetsov S (2012) Micro-inertia effects in nonlinear heterogeneous media. Int J Numer Methods Eng 91(13):1406–1426. doi:10.1002/nme.4322

    Article  MathSciNet  Google Scholar 

  22. Fish J, Filonova V, Fafalis D (2014) Computational continua revisited. Int J Numer Methods Eng. doi:10.1002/nme.4793

  23. Hegemier G (1978) Mixture theories with microstructure for wave propagation and diffusion in composite materials. Solid Mech Arch 3(1):33–71

    Google Scholar 

  24. Hou T, Wu XH (1997) A multiscale finite element method for elliptic problems in composite materials and porous media. J Comput Phys 134(1):169–189. doi:10.1006/jcph.1997.5682

    Article  MATH  MathSciNet  Google Scholar 

  25. Hughes T, Feijo G, Mazzei L, Quincy JB (1998) The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166(1–2):3–24

    Article  MATH  Google Scholar 

  26. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis, dover civil and mechanical engineering. Dover Publications, Mineola

    Google Scholar 

  27. Hui T, Oskay C (2013) A nonlocal homogenization model for wave dispersion in dissipative composite materials. Int J Solids Struct 50(1):38–48. doi:10.1016/j.ijsolstr.2012.09.007

    Article  Google Scholar 

  28. Hui T, Oskay C (2014) A high order homogenization model for transient dynamics of heterogeneous media including micro-inertia effects. Comput Methods Appl Mech Eng 273:181–203. doi:10.1016/j.cma.2014.01.028

    Article  MATH  MathSciNet  Google Scholar 

  29. Hui T, Oskay C (2015) Laplace-domain, high-order homogenization for transient dynamic response of viscoelastic composites. Int J Numer Methods Eng. doi:10.1002/nme.4793, doi:10.1002/nme.4916

  30. Kanaun S, Levin V (2003) Self-consistent methods in the problem of axial elastic shear wave propagation through fiber composites. Arch Appl Mech 73(1–2):105–130. doi:10.1007/s00419-002-0250-9

    MATH  Google Scholar 

  31. Li A, Waisman H, Fish J (2006) A space-time multiscale method for molecular dynamics simulations of biomolecules. Int J Multiscale Comput Eng 4(5–6):791–801. doi:10.1615/IntJMultCompEng.v4.i5-6.120

    Article  Google Scholar 

  32. Li A, Li R, Fish J (2008) Generalized mathematical homogenization: from theory to practice. Comput Methods Appl Mech Eng 197(41–42):3225–3248. doi:10.1016/j.cma.2007.12.002

    Article  MATH  MathSciNet  Google Scholar 

  33. Liu Z, Zhang X, Mao Y, Zhu Y, Yang Z, Chan C, Sheng P (2000) Locally resonant sonic materials. Science 289(5485):1734–1736. doi:10.1126/science.289.5485.1734

    Article  Google Scholar 

  34. Maxwell JC (1874) van der waals on the continuity of the gaseous and liquid states. Nature 10:477–80

    Article  Google Scholar 

  35. Nayfeh AH, Gurtman G (1973) Continuum approach to the propagation of shear waves in laminated wave guides. ASME Pap (73 -APMW-12)

  36. Nemat-Nasser S, Srivastava A (2011) Overall dynamic constitutive relations of layered elastic composites. J Mech Phys Solids 59(10):1953–1965. doi:10.1016/j.jmps.2011.07.008

    Article  MATH  MathSciNet  Google Scholar 

  37. Nolde E, Rc Craster, Kaplunov J (2011) High frequency homogenization for structural mechanics. J Mech Phys Solids 59(3):651–671. doi:10.1016/j.jmps.2010.12.004

    Article  MATH  MathSciNet  Google Scholar 

  38. Pham K, Kouznetsova V, Geers M (2013) Transient computational homogenization for heterogeneous materials under dynamic excitation. J Mech Phys Solids 61(11):2125–2146. doi:10.1016/j.jmps.2013.07.005

    Article  MathSciNet  Google Scholar 

  39. Sabina F, Willis J (1988) A simple self-consistent analysis of wave propagation in particulate composites. Wave Motion 10(2):127–142. doi:10.1016/0165-2125(88)90038-8

    Article  MATH  Google Scholar 

  40. Santosa F, Symes WW (1991) A dispersive effective medium for wave propagation in periodic composites. SIAM J Appl Math 51(4):984–1005

    Article  MATH  MathSciNet  Google Scholar 

  41. Waisman H, Fish J (2006) A space-time multilevel method for molecular dynamics simulations. Comput Methods Appl Mech Eng 195(44–47):6542–6559. doi:10.1016/j.cma.2006.02.006

    Article  MATH  MathSciNet  Google Scholar 

  42. Weinan E, Engquist B, Huang Z (2003) Heterogeneous multiscale method: a general methodology for multiscale modeling. Phys Rev B 67(9):921011–921014

  43. Wing KL (1991) Multi-scale finite element methods for structural dynamics. American Institute of Aeronautics and Astronautics AIAA-91-1057-CP, pp 2510–2514

  44. Zohdi TI, Wriggers P (2005) An introduction to computational micromechanics, vol 20. Lecture notes in applied and computational mechanics. Springer, Berlin. doi:10.1007/978-3-540-32360-0

Download references

Acknowledgments

The ONR Grant N00014-12-1-0558 to Columbia University is gratefully acknowledged.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jacob Fish.

Appendix: Monolithic form of the coupled system

Appendix: Monolithic form of the coupled system

The coupled problem consists of solving at every time step the C-problem (26) and the F-problem (23) for all the unit cells in the domain. In a monolithic matrix form this can be expressed as follows:

$$\begin{aligned} \left[ {\begin{array}{cc} {{\mathbf{M }^F}}&{}{{\mathbf{M }^{FC}}} \\ {{\mathbf{M }^{CF}}}&{}{{\mathbf{M }^C}} \end{array}} \right] \left\{ {\begin{array}{c} {{{{\ddot{\mathbf{d }}}}^{\left( 1 \right) }}} \\ {{{{\ddot{\mathbf{d }}}}^C}} \end{array}} \right\} + \left[ {\begin{array}{cc} {{\mathbf{K }^F}}&{}{{\mathbf{K }^{FC}}} \\ {{\mathbf{K }^{CF}}}&{}{{\mathbf{K }^C}} \end{array}} \right] \left\{ {\begin{array}{c} {{\mathbf{d }^{\left( 1 \right) }}} \\ {{\mathbf{d }^C}} \end{array}} \right\} = \left\{ {\begin{array}{c} \mathbf{F }^{(1)} \\ \mathbf{F }^{C} \end{array}} \right\} \end{aligned}$$
(70)

For simplicity, the following matrices can been defined:

$$\begin{aligned} \mathbf M&= \left[ {\begin{array}{cc} {{\mathbf{M }^F}}&{}{{\mathbf{M }^{FC}}} \\ {{\mathbf{M }^{CF}}}&{}{{\mathbf{M }^C}} \end{array}} \right] \end{aligned}$$
(71a)
$$\begin{aligned} \mathbf K&= \left[ {\begin{array}{cc} {{\mathbf{K }^F}}&{}{{\mathbf{K }^{FC}}} \\ {{\mathbf{K }^{CF}}}&{}{{\mathbf{K }^C}} \end{array}} \right] \end{aligned}$$
(71b)
$$\begin{aligned} \mathbf{ d }&= \left\{ {\begin{array}{c} {{\mathbf{d }^{(1)}}} \\ {{\mathbf{d }^C}} \end{array}} \right\} \end{aligned}$$
(71c)
$$\begin{aligned} \mathbf F&= \left\{ {\begin{array}{c} \mathbf{F }^{(1)} \\ \mathbf{F }^{C} \end{array}} \right\} \end{aligned}$$
(71d)

so that it can be written in a compact monolithic form:

$$\begin{aligned} \mathbf{M } {\ddot{\mathbf{d }}} + \mathbf K \mathbf{ d } = \mathbf F \end{aligned}$$
(72)

The assembled matrices are of the following form:

(73a)
(73b)
(73c)
(74)

\({\mathbf{M }^{C}_{e}}, {\mathbf{K }^{C}_{e}}\) from (27)

(75a)
(75b)
(76a)
(76b)
(77a)
(77b)
(78a)
(78b)

where, operator constructs block diagonal matrices, and operator assembles over the coarse-scale and unit cell DOFs.

Similarly for the coupling matrices \({\mathbf{M }^{FC}}, {\mathbf{K }^{FC}}\)

$$\begin{aligned} {\mathbf{M }^{FC}_{e}} = \left[ {\begin{array}{*{20}{c}} {\mathbf{M }^{FC}_1} \\ {\mathbf{M }^{FC}_{2}} \\ \cdots \\ {\mathbf{M }^{FC}_{{\hat{N}}_e}} \end{array}} \right] \end{aligned}$$
(79a)
(79b)
$$\begin{aligned} \mathbf K ^{FC}_{e} = \left[ {\begin{array}{*{20}{c}} {\mathbf{K }^{FC}_1} \\ {\mathbf{K }^{FC}_{2}} \\ \cdots \\ {\mathbf{K }^{FC}_{{\hat{N}}_e}} \end{array}} \right] \end{aligned}$$
(80a)
(80b)

Remark

It is emphasized that fine-scale matrices \(\mathbf{M }^F_I, \mathbf{K }^F_I\) are decoupled to each other. So do fine-scale displacements, which are coupled only to the coarse-scale displacements of the coarse-scale element they belong to. However, coupling matrices \({\mathbf{M }^{FC}_I}, {\mathbf{K }^{FC}_I},{\mathbf{M }^{CF}_I}, {\mathbf{K }^{CF}_I}\) of different unit cells are coupled to each other through the common coarse-scale nodes.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fafalis, D., Fish, J. Computational aspects of dispersive computational continua for elastic heterogeneous media. Comput Mech 56, 931–946 (2015). https://doi.org/10.1007/s00466-015-1211-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-015-1211-9

Keywords

Navigation