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Microstructure-informed reduced modes synthesized with Wang tiles and the Generalized Finite Element Method

Abstract

A recently introduced representation by a set of Wang tiles—a generalization of the traditional Periodic Unit Cell-based approach—serves as a reduced geometrical model for materials with stochastic heterogeneous microstructure, enabling an efficient synthesis of microstructural realizations. To facilitate macroscopic analyses with a fully resolved microstructure generated with Wang tiles, we develop a reduced order modelling scheme utilizing pre-computed characteristic features of the tiles. In the offline phase, inspired by computational homogenization, we extract continuous fluctuation fields from the compressed microstructural representation as responses to generalized loading represented by the first- and second-order macroscopic gradients. In the online phase, using the ansatz of the generalized finite element method, we combine these fields with a coarse finite element discretization to create microstructure-informed reduced modes specific for a given macroscopic problem. Considering a two-dimensional scalar elliptic problem, we demonstrate that our scheme delivers less than 3% error in both the relative \(L_2\) and energy norms with only 0.01% of the unknowns when compared to the fully resolved problem. Accuracy can be further improved by locally refining the macroscopic discretization and/or employing more pre-computed fluctuation fields. Finally, unlike standard snapshot-based reduced-order approaches, our scheme handles significant changes in the macroscopic geometry or loading without the need for recalculating the offline phase, because the fluctuation fields are extracted without any prior knowledge of the macroscopic problem.

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Notes

  1. 1.

    In the case of zero fluctuations prescribed at tile boundaries, i.e. Eq. (18), the above described treatment is only formal. In practice, solving Eq. (25) is skipped, and only the interior DOFs are sought for because no information is effectively communicated across individual tiles.

  2. 2.

    https://www.mathworks.com/help/matlab/call-mex-file-functions.html.

References

  1. 1.

    Amsallem D, Zahr MJ, Farhat C (2012) Nonlinear model order reduction based on local reduced-order bases. Int J Numer Meth Eng 92(10):891–916. https://doi.org/10.1002/nme.4371

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    An SS, Kim T, James DL (2008) Optimizing cubature for efficient integration of subspace deformations. ACM Trans Graph 27(5):1. https://doi.org/10.1145/1409060.1409118

    Article  Google Scholar 

  3. 3.

    Astrid P, Weiland S, Willcox K, Backx T (2008) Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control 53(10):2237–2251. https://doi.org/10.1109/TAC.2008.2006102

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Balzani D, Scheunemann L, Brands D, Schröder J (2014) Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations. Comput Mech 54(5):1269–1284. https://doi.org/10.1007/s00466-014-1057-6

    Article  MATH  Google Scholar 

  5. 5.

    Barbič J, James DL (2005) Real-Time subspace integration for St. Venant-Kirchhoff deformable models. ACM Trans Graph 24(3):982–990. https://doi.org/10.1145/1073204.1073300

    Article  Google Scholar 

  6. 6.

    Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modeling. Modell Simul Mater Sci Eng 17(4):043001. https://doi.org/10.1088/0965-0393/17/4/043001

    Article  Google Scholar 

  7. 7.

    Bolzon G, Buljak V (2011) An effective computational tool for parametric studies and identification problems in materials mechanics. Comput Mech 48(6):675–687. https://doi.org/10.1007/s00466-011-0611-8

    Article  MATH  Google Scholar 

  8. 8.

    Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764. https://doi.org/10.1137/090766498

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Cohen MF, Shade J, Hiller S, Deussen O (2003) Wang tiles for image and texture generation. ACM Trans Graph 22(3):287–294. https://doi.org/10.1145/882262.882265

    Article  Google Scholar 

  10. 10.

    Doškář M, Novák J (2016) A jigsaw puzzle framework for homogenization of high porosity foams. Comput Struct 166:33–41. https://doi.org/10.1016/j.compstruc.2016.01.003

    Article  Google Scholar 

  11. 11.

    Doškář M, Novák J, Zeman J (2014) Aperiodic compression and reconstruction of real-world material systems based on Wang tiles. Phys Rev E 90(6):062118. https://doi.org/10.1103/PhysRevE.90.062118

    Article  Google Scholar 

  12. 12.

    Doškář M, Zeman J, Jarušková D, Novák J (2018) Wang tiling aided statistical determination of the representative volume element size of random heterogeneous materials. Eur J Mech A Solids 70:280–295. https://doi.org/10.1016/j.euromechsol.2017.12.002

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Doškář M, Zeman J, Rypl D, Novák J (2020) Level-set based design of wang tiles for modelling complex microstructures. Comput Aided Des 123:102827. https://doi.org/10.1016/j.cad.2020.102827

    MathSciNet  Article  Google Scholar 

  14. 14.

    Duarte C, Kim DJ (2008) Analysis and applications of a generalized finite element method with global–local enrichment functions. Comput Methods Appl Mech Eng 197(6–8):487–504. https://doi.org/10.1016/j.cma.2007.08.017

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Efendiev Y, Galvis J, Hou TY (2013) Generalized multiscale finite element methods (GMsFEM). J Comput Phys 251:116–135. https://doi.org/10.1016/j.jcp.2013.04.045

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Eftang JL, Stamm B (2012) Parameter multi-domain ‘hp’ empirical interpolation. Int J Numer Meth Eng 90(4):412–428. https://doi.org/10.1002/nme.3327

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Feyel F, Chaboche JL (2000) FE\(^{2}\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183(3–4):309–330. https://doi.org/10.1016/S0045-7825(99)00224-8

    Article  MATH  Google Scholar 

  18. 18.

    Fish J, Yuan Z (2005) Multiscale enrichment based on partition of unity. Int J Numer Meth Eng 62(10):1341–1359. https://doi.org/10.1002/nme.1230

    Article  MATH  Google Scholar 

  19. 19.

    Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Meth Eng 84(3):253–304. https://doi.org/10.1002/nme.2914

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Fritzen F, Xia L, Leuschner M, Breitkopf P (2016) Topology optimization of multiscale elastoviscoplastic structures. Int J Numer Meth Eng 106(6):430–453. https://doi.org/10.1002/nme.5122

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Fritzen F, Haasdonk B, Ryckelynck D, Schöps S (2018) An algorithmic comparison of the hyper-reduction and the discrete empirical interpolation method for a nonlinear thermal problem. Math Comput Appl 23(1):8. https://doi.org/10.3390/mca23010008

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Geers MGD, Kouznetsova VG, Matouš K, Yvonnet J (2017) Homogenization methods and multiscale modeling: nonlinear problems. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, 2nd edn. Wiley, Chichester, pp 1–34. https://doi.org/10.1002/9781119176817.ecm107

    Chapter  Google Scholar 

  23. 23.

    Goury O, Amsallem D, Bordas SPA, Liu WK, Kerfriden P (2016) Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to Bayesian optimization. Comput Mech 58(2):213–234. https://doi.org/10.1007/s00466-016-1290-2

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Grünbaum B, Shephard GC (2016) Tilings and patterns, 2nd edn. Dover Publications Inc, Mineola, New York

    MATH  Google Scholar 

  25. 25.

    Guennebaud G, Jacob B et al. (2010) Eigen v3. http://eigen.tuxfamily.org

  26. 26.

    Harmon D, Zorin D (2013) Subspace integration with local deformations. ACM Trans Graph 32(4):1. https://doi.org/10.1145/2461912.2461922

    Article  MATH  Google Scholar 

  27. 27.

    Hiller S, Deussen O, Keller A (2001) Tiled blue noise samples. In: Proceedings of the Vision Modeling and Visualization Conference 2001, Aka GmbH, Stuttgart, Germany, VMV ’01, pp 265–272

  28. 28.

    Ibáñez R, Abisset-Chavanne E, Chinesta F, Huerta A, Cueto E (2019) A local, multiple proper generalized decomposition based on the partition of unity. Int J Numer Methods Eng. https://doi.org/10.1002/nme.6128

    MathSciNet  Article  Google Scholar 

  29. 29.

    Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40(13–14):3647–3679. https://doi.org/10.1016/S0020-7683(03)00143-4

    Article  MATH  Google Scholar 

  30. 30.

    Kerfriden P, Goury O, Rabczuk T, Bordas S (2013) A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics. Comput Methods Appl Mech Eng 256:169–188. https://doi.org/10.1016/j.cma.2012.12.004

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Kerschen G, Golinval JC, Vakakis AF, Bergman LA (2005) The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn 41(1–3):147–169. https://doi.org/10.1007/s11071-005-2803-2

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Kim T, James DL (2011) Physics-based character skinning using multi-domain subspace deformations. In: SCA ’11: Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, ACM Press, p 63. https://doi.org/10.1145/2019406.2019415

  33. 33.

    Kopf J, Cohen-Or D, Deussen O, Lischinski D (2006) Recursive Wang tiles for real-time blue noise. ACM Trans Graph 25(3):509. https://doi.org/10.1145/1141911.1141916

    Article  Google Scholar 

  34. 34.

    Kouznetsova V, Geers MG, Brekelmans WM (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Meth Eng 54(8):1235–1260

    Article  Google Scholar 

  35. 35.

    Kouznetsova V, Geers M, Brekelmans W (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193(48–51):5525–5550. https://doi.org/10.1016/j.cma.2003.12.073

    Article  MATH  Google Scholar 

  36. 36.

    Kunc O, Fritzen F (2019) Finite strain homogenization using a reduced basis and efficient sampling. Math Comput Appl 24(2):56. https://doi.org/10.3390/mca24020056

    MathSciNet  Article  Google Scholar 

  37. 37.

    Lagae A, Dutré P (2006) An alternative for Wang tiles: colored edges versus colored corners. ACM Trans Graph 25(4):1442–1459. https://doi.org/10.1145/1183287.1183296

    Article  Google Scholar 

  38. 38.

    Le MV, Yvonnet J, Feld N, Detrez F (2020a) The coarse mesh condensation multiscale method for parallel computation of heterogeneous linear structures without scale separation. Comput Methods Appl Mech Eng 363:112877. https://doi.org/10.1016/j.cma.2020.112877

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Le MV, Yvonnet J, Feld N, Detrez F (2020b) Full-field elastic simulations for image-based heterogeneous structures with a coarse mesh condensation multiscale method. Int J Multiscale Comput Eng. https://doi.org/10.1615/IntJMultCompEng.2020034828

    Article  MATH  Google Scholar 

  40. 40.

    Lee H, Brandyberry M, Tudor A, Matouš K (2009) Three-dimensional reconstruction of statistically optimal unit cells of polydisperse particulate composites from microtomography. Phys Rev E 80(6):061301. https://doi.org/10.1103/PhysRevE.80.061301

    Article  Google Scholar 

  41. 41.

    Liu X, Shapiro V (2015) Random heterogeneous materials via texture synthesis. Comput Mater Sci 99:177–189. https://doi.org/10.1016/j.commatsci.2014.12.017

    Article  Google Scholar 

  42. 42.

    Matache A, Babuška I, Schwab C (2000) Generalized p-FEM in homogenization. Numer Math 86(2):319–375. https://doi.org/10.1007/PL00005409

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Melenk J, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1–4):289–314. https://doi.org/10.1016/S0045-7825(96)01087-0

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Niroomandi S, Alfaro I, Cueto E, Chinesta F (2012a) Accounting for large deformations in real-time simulations of soft tissues based on reduced-order models. Comput Methods Programs Biomed 105(1):1–12. https://doi.org/10.1016/j.cmpb.2010.06.012

    Article  Google Scholar 

  45. 45.

    Niroomandi S, Alfaro I, González D, Cueto E, Chinesta F (2012b) Real-time simulation of surgery by reduced-order modeling and X-FEM techniques. Int J Numer Methods Biomed Eng 28(5):574–588. https://doi.org/10.1002/cnm.1491

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    Novák J, Kaczmarczyk L, Grassl P, Zeman J, Pearce CJ (2012a) A micromechanics-enhanced finite element formulation for modelling heterogeneous materials. Comput Methods Appl Mech Eng 201–204:53–64. https://doi.org/10.1016/j.cma.2011.09.003

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Novák J, Kučerová A, Zeman J (2012b) Compressing random microstructures via stochastic Wang tilings. Phys Rev E 86(4):040104(R). https://doi.org/10.1103/PhysRevE.86.040104

    Article  Google Scholar 

  48. 48.

    Novák J, Kučerová A, Zeman J (2013) Microstructural enrichment functions based on stochastic Wang tilings. Modell Simul Mater Sci Eng 21(2):025014. https://doi.org/10.1088/0965-0393/21/2/025014

    Article  Google Scholar 

  49. 49.

    Oliver J, Caicedo M, Huespe A, Hernández J, Roubin E (2017) Reduced order modeling strategies for computational multiscale fracture. Comput Methods Appl Mech Eng 313:560–595. https://doi.org/10.1016/j.cma.2016.09.039

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Peherstorfer B, Butnaru D, Willcox K, Bungartz HJ (2014) Localized discrete empirical interpolation method. SIAM J Sci Comput 36(1):A168–A192. https://doi.org/10.1137/130924408

    MathSciNet  Article  MATH  Google Scholar 

  51. 51.

    Plews J, Duarte C (2015) Bridging multiple structural scales with a generalized finite element method. Int J Numer Meth Eng 102(3–4):180–201. https://doi.org/10.1002/nme.4703

    MathSciNet  Article  MATH  Google Scholar 

  52. 52.

    Radermacher A, Reese S (2013) A comparison of projection-based model reduction concepts in the context of nonlinear biomechanics. Arch Appl Mech 83(8):1193–1213. https://doi.org/10.1007/s00419-013-0742-9

    Article  MATH  Google Scholar 

  53. 53.

    Radermacher A, Reese S (2014) Model reduction in elastoplasticity: proper orthogonal decomposition combined with adaptive sub-structuring. Comput Mech 54(3):677–687. https://doi.org/10.1007/s00466-014-1020-6

    MathSciNet  Article  MATH  Google Scholar 

  54. 54.

    Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202(1):346–366. https://doi.org/10.1016/j.jcp.2004.07.015

    MathSciNet  Article  MATH  Google Scholar 

  55. 55.

    Ryckelynck D (2009) Hyper-reduction of mechanical models involving internal variables. Int J Numer Meth Eng 77(1):75–89. https://doi.org/10.1002/nme.2406

    MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    Sedgewick R (2002) Algorithms in C++, Part 5: Graph Algorithms, 3rd edn. Addison-Wesley, Boston

    Google Scholar 

  57. 57.

    Sibley PG, Montgomery P, Marai GE (2004) Wang cubes for video synthesis and geometry placement. In: Proceeding SIGGRAPH ’04 ACM SIGGRAPH 2004 Posters. ACM Press, p 20, https://doi.org/10.1145/1186415.1186439

  58. 58.

    Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190(32–33):4081–4193. https://doi.org/10.1016/S0045-7825(01)00188-8

    MathSciNet  Article  MATH  Google Scholar 

  59. 59.

    Strouboulis T, Zhang L, Babuška I (2003) Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids. Comput Methods Appl Mech Eng 192(28–30):3109–3161. https://doi.org/10.1016/S0045-7825(03)00347-5

    MathSciNet  Article  MATH  Google Scholar 

  60. 60.

    Tyburec M, Zeman J, Doškář M, Kružík M, Lepš M (2021) Modular-topology optimization with Wang tilings: an application to truss structures. Struct Multidiscip Optim 63(3):1099–1117. https://doi.org/10.1007/s00158-020-02744-8

    MathSciNet  Article  Google Scholar 

  61. 61.

    Wang H (1961) Proving theorems by pattern recognition—II. Bell Syst Tech J 40(1):1–41. https://doi.org/10.1002/j.1538-7305.1961.tb03975.x

    Article  Google Scholar 

  62. 62.

    Xia L, Breitkopf P (2014) A reduced multiscale model for nonlinear structural topology optimization. Comput Methods Appl Mech Eng 280:117–134. https://doi.org/10.1016/j.cma.2014.07.024

    MathSciNet  Article  MATH  Google Scholar 

  63. 63.

    Yang M, Nagarajan A, Liang B, Soghrati S (2018) New algorithms for virtual reconstruction of heterogeneous microstructures. Comput Methods Appl Mech Eng 338:275–298. https://doi.org/10.1016/j.cma.2018.04.030

    MathSciNet  Article  MATH  Google Scholar 

  64. 64.

    Yvonnet J (2019) Computational homogenization of heterogeneous materials with finite elements, solid mechanics and its applications, vol 258. Springer, Berlin. https://doi.org/10.1007/978-3-030-18383-7

    Book  MATH  Google Scholar 

  65. 65.

    Yvonnet J, He QC (2007) The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys 223(1):341–368. https://doi.org/10.1016/j.jcp.2006.09.019

    MathSciNet  Article  MATH  Google Scholar 

  66. 66.

    Zeman J, Šejnoha M (2007) From random microstructures to representative volume elements. Modell Simul Mater Sci Eng 15(4):S325–S335. https://doi.org/10.1088/0965-0393/15/4/S01

    Article  Google Scholar 

  67. 67.

    Zhang X, Kim YJ (2008) Efficient texture synthesis using strict Wang Tiles. Graph Models 70(3):43–56. https://doi.org/10.1016/j.gmod.2007.10.002

    Article  Google Scholar 

  68. 68.

    Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method: its basis and fundamentals, 7th edn. Elsevier, Butterworth-Heinemann, Amsterdam

    MATH  Google Scholar 

  69. 69.

    Zohdi T, Wriggers P (1999) A domain decomposition method for bodies with heterogeneous microstructure based on material regularization. Int J Solids Struct 36(17):2507–2525. https://doi.org/10.1016/S0020-7683(98)00124-3

    MathSciNet  Article  MATH  Google Scholar 

  70. 70.

    Zohdi T, Wriggers P, Huet C (2001) A method of substructuring large-scale computational micromechanical problems. Comput Methods Appl Mech Eng 190(43–44):5639–5656. https://doi.org/10.1016/S0045-7825(01)00189-X

    Article  MATH  Google Scholar 

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Acknowledgements

This research was funded by the Czech Science Foundation, Project No. 19-26143X. Martin Doškář also gratefully acknowledges the support from Fulbright Commission Czech Republic that funded his research stay at the University of California, San Diego, from 2016 to 2017. We thank Stephanie Krueger for a critical review and proof-reading of the initial versions of this manuscript. We also appreciate constructive comments by anonymous reviewers.

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Appendices

Expressions for stiffness matrix and load vector

Assuming the standard Finite Element approximation of the solution in the form of a piecewise polynomial, recall Eq. (7), individual entries of the stiffness matrix \(\mathsf {K}^{\mathcal {T}}\), which corresponds to the Hessian of \({\tilde{\varPi }}\left( {\varvec{G}}^{1}\!,\mathbf {G}^{2}\!,{\hat{\theta }}\right) \) with fixed \({\varvec{G}}^{1}\!\) and \(\mathbf {G}^{2}\!\), follow from the standard expression

$$\begin{aligned} K^{\mathcal {T}}_{mn} = \int _{\varOmega ^{\mathcal {T}}} {\varvec{\nabla }} N ^{\mathcal {T}}_{m}({\varvec{x}})\cdot \mathbf {K}({\varvec{x}})\cdot {\varvec{\nabla }}N^{\mathcal {T}}_{n} \,{\mathrm d}{\varvec{x}} \,. \end{aligned}$$
(36)

Analogously, components of load vector \(\mathsf {f}^{\mathcal {T}}\left( {\varvec{G}}^{1}\!, \mathbf {G}^{2}\!\right) \) from Eq. (8) are obtained by plugging the assumed solution (5) into the stationary conditions of \({\tilde{\varPi }}\left( {\varvec{G}}^{1}\!,\mathbf {G}^{2}\!,{\hat{\theta }}\right) \) with respect to the sought-for coefficients \(u^{\mathcal {T}}_{m}\)

$$\begin{aligned} f^{\mathcal {T}}_{m}\left( {\varvec{G}}^{1}\!, \mathbf {G}^{2}\!\right) = -\int _{\varOmega ^{\mathcal {T}}} {\varvec{\nabla }} N ^{\mathcal {T}}_{m}({\varvec{x}})\cdot \mathbf {K}({\varvec{x}})\cdot ({\varvec{G}}^{1}\!+ \mathbf {G}^{2}\!\cdot {\varvec{x}}) \,{\mathrm d}{\varvec{x}} \,.\nonumber \\ \end{aligned}$$
(37)

The integrals are computed element-wise, using the Gauss numerical quadrature rule and the standard finite element procedures [68].

Unique vertex numbering

To generate fluctuation fields that are by construction continuous, boundary degrees of freedom (DOFs) must be enumerated consistently, i.e. corresponding nodes at the edges with the same code within different tiles should be assigned the same number. While the enumeration is straightforward for edge-related DOFs, it does not hold for vertex DOFs. Assigning the same number to all vertex DOFs is not always correct because—depending on the code distribution within the set—it may happen that certain vertices will never coincide during the tile assembly, thus there might be separate vertex group with distinct DOFs. This typically happens for tile sets derived from the vertex-defined Wang tiles [37].

Inspired by the pragmatic question of where to copy a particle that intersects more than one edge during microstructural generation [13], we devised an algorithm that is capable of identifying the separate vertex groups. First, we construct an undirected graph from the provided tile set definition such that (i) each node of the graph corresponds to one half of each edge code (i.e. there is a left and right part for each horizontal and top and bottom part for each vertical code) and (ii) arcs between the graph nodes are obtained from the individual vertices of each tile.

Next, using the Depth-First Search algorithm [56, Section 18.2], we identify the connected components of the graph. If several sub-graphs are identified, the arcs corresponding to this group yield vertices that belong to a distinct group and only these should be assigned the same number. For more details, including illustrations for several tile sets, readers are referred to [13, Section 2.3].

Discrete form of constraints

For completeness, implementation details of the individual constraint matrices from Eq. (24) are provided next. We assume the standard isoparametric finite elements [68]. The boundary of each domain is consistently discretized by restricting planar finite element discretization to the tiles’ boundary, e.g. domain discretization by linear triangles yields linear line elements \({\bar{e}}\) on the boundary. Since the tile domains are square and aligned with coordinate axes, the outer normal \({\varvec{n}} = \begin{bmatrix} n_{x}&n_{y} \end{bmatrix}^\mathsf{T}\) along individual edge elements is constant, which simplifies the following expressions.

First, the average value of \(\tilde{ \theta }\) along all boundaries of all tiles, appearing in Eq. (16), is obtained through

$$\begin{aligned} \begin{aligned} \int _{\partial \varOmega ^{\mathcal {S}}} \tilde{ \theta }\,{\mathrm d}{\varvec{x}}&= \sum _{\mathcal {T}\in \mathcal {S}} \int _{\partial \varOmega ^{\mathcal {T}}} \tilde{ \theta }\,{\mathrm d}{\varvec{x}} \\&\approx \sum _{\mathcal {T}\in \mathcal {S}} \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}({\varvec{x}})\mathsf {u}^{{\bar{e}},\mathcal {T}} \,{\mathrm d}{\varvec{x}} \\&= \sum _{\mathcal {T}\in \mathcal {S}} \left( \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}({\varvec{x}})\,{\mathrm d}{\varvec{x}} \, \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {u}^{\mathcal {T}}_{b} \\&= \underbrace{ \left\{ \sum _{\mathcal {T}\in \mathcal {S}} \left( \sum _{{\bar{e}}} \left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {L}^{\mathcal {T}} \right\} }_{\mathsf {C}_{0}} \mathsf {u}^{\text {s}} \,, \end{aligned} \end{aligned}$$
(38)

where \(\bar{\mathsf {N}}\) is a row vector of shape functions pertinent to the vertices of element \({\bar{e}}\), \(\left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}}\) denotes its integrated counterpart using a suitable quadrature rule, \(\mathsf {L}^{{\bar{e}},\mathcal {T}}\) maps boundary DOFs of tile \(\mathcal {T}\) onto nodal unknowns of boundary element \({\bar{e}}\) such that \(\mathsf {u}^{{\bar{e}},\mathcal {T}} = \mathsf {L}^{{\bar{e}},\mathcal {T}} \mathsf {u}^{\mathcal {T}}_{b}\), and \(\mathsf {L}^{\mathcal {T}}\) is the localization matrix introduced in Sect. 3.1.

Proceeding with Eq. (17), the boundary integral related to the average of the first-order gradient of each tile is approximated as

$$\begin{aligned} \begin{aligned} \int _{\partial \varOmega ^{\mathcal {T}}} \tilde{ \theta }{\varvec{n}} \,{\mathrm d}s&\approx \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} \begin{bmatrix} n^{{\bar{e}}}_{x} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\\ n^{{\bar{e}}}_{y} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\end{bmatrix} \mathsf {u}^{{\bar{e}},\mathcal {T}} \,{\mathrm d}{\varvec{x}} \\&= \left( \sum _{{\bar{e}}} \begin{bmatrix} n^{{\bar{e}}}_{x}\int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\,{\mathrm d}{\varvec{x}} \\ n^{{\bar{e}}}_{y}\int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\,{\mathrm d}{\varvec{x}} \end{bmatrix} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {u}^{\mathcal {T}}_{b} \\&= \underbrace{ \left( \sum _{{\bar{e}}} \begin{bmatrix} n^{{\bar{e}}}_{x} \left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \\ n^{{\bar{e}}}_{y} \left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \end{bmatrix} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {L}^{\mathcal {T}} }_{\mathsf {C}^{\mathcal {T}}_{\text {I}}} \mathsf {u}^{\mathcal {S}} \,. \end{aligned} \end{aligned}$$
(39)

Depending whether the average first-order gradient is set to vanish for each tile individually, Eq. (19), or for the whole set only, Eq. (20), the corresponding \(\mathsf {C}_{\text {I}}\) takes either the form

$$\begin{aligned} \mathsf {C}_{\text {I}} = \begin{bmatrix} \mathsf {C}^{\mathcal {T}_{1}}_{\text {I}} \\ \vdots \\ \mathsf {C}^{\mathcal {T}_{16}}_{\text {I}} \end{bmatrix} \end{aligned}$$
(40)

or

$$\begin{aligned} \mathsf {C}_{\text {I}} = \sum _{\mathcal {T}\in \mathcal {S}} \mathsf {C}^{\mathcal {T}}_{\text {I}} \,. \end{aligned}$$
(41)

Analogously, a similar approximation is used also for the second-order gradient constraints. Note that the second-order gradient is a symmetric \(2\times 2\) tensor; hence, it constitutes only three independent scalar constraints, which we represent with a vector using Voigt notation. Starting with the constraints (21) posed on an individual tile and with \({\varvec{x}} = \begin{bmatrix}x&y\end{bmatrix}^\mathsf{T}\), we obtain

$$\begin{aligned} \begin{aligned} \int _{\partial \varOmega ^{\mathcal {T}}}&\tilde{ \theta }\left( {\varvec{x}}\otimes {\varvec{n}} + {\varvec{n}} \otimes {\varvec{x}}\right) \,{\mathrm d}s \\&\approx \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} \begin{bmatrix} 2 n^{{\bar{e}}}_{x} x {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\\ 2 n^{{\bar{e}}}_{y} y {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\\ (n^{{\bar{e}}}_{x} y + n^{{\bar{e}}}_{y} x) {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\end{bmatrix} \mathsf {u}^{{\bar{e}},\mathcal {T}} \,{\mathrm d}{\varvec{x}} \\&= \underbrace{ \left\{ \sum _{{\bar{e}}} \left( \begin{bmatrix} 2 n^{{\bar{e}}}_{x} \left\langle \mathsf {x} {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \\ 2 n^{{\bar{e}}}_{y} \left\langle \mathsf {y} {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \\ n^{{\bar{e}}}_{x} \left\langle \mathsf {y} {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} + n^{{\bar{e}}}_{y} \left\langle \mathsf {y}{\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \end{bmatrix} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {L}^{\mathcal {T}} \right\} }_{\mathsf {C}^{\mathcal {T}}_{\text {II}}} \mathsf {u}^{\mathcal {S}} \,, \end{aligned} \end{aligned}$$
(42)

with \(\left\langle \mathsf {x}{\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}}\) and \(\left\langle \mathsf {y}{\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}}\) denoting the corresponding quantities integrated over element \({\bar{e}}\). Again, depending whether the constraint is posed on individual tiles, Eq. (22), or the set as a whole, Eq. (23), the tile constraint matrices \(\mathsf {C}^{\mathcal {T}}_{\text {II}}\) are either stacked,

$$\begin{aligned} \mathsf {C}_{\text {II}} = \begin{bmatrix} \mathsf {C}^{\mathcal {T}_{1}}_{\text {II}} \\ \vdots \\ \mathsf {C}^{\mathcal {T}_{16}}_{\text {II}} \end{bmatrix} \,, \end{aligned}$$
(43)

or summed,

$$\begin{aligned} \mathsf {C}_{\text {II}} = \sum _{\mathcal {T}\in \mathcal {S}} \mathsf {C}^{\mathcal {T}}_{\text {II}} \,. \end{aligned}$$
(44)

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Doškář, M., Zeman, J., Krysl, P. et al. Microstructure-informed reduced modes synthesized with Wang tiles and the Generalized Finite Element Method. Comput Mech 68, 233–253 (2021). https://doi.org/10.1007/s00466-021-02028-y

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Keywords

  • Wang tiling
  • Microstructure-informed modes
  • Reduced order modelling
  • Heterogeneous materials