An adaptive wavelet-based collocation method for solving multiscale problems in continuum mechanics
Tóm tắt
Computational multiscale methods are highly sophisticated numerical approaches to predict the constitutive response of heterogeneous materials from their underlying microstructures. However, the quality of the prediction intrinsically relies on an accurate representation of the microscale morphology and its individual constituents, which makes these formulations computationally demanding. Against this background, the applicability of an adaptive wavelet-based collocation approach is studied in this contribution. It is shown that the Hill–Mandel energy equivalence condition can naturally be accounted for in the wavelet basis, (discrete) wavelet-based scale-bridging relations are derived, and a wavelet-based mapping algorithm for internal variables is proposed. The characteristic properties of the formulation are then discussed by an in-depth analysis of elementary one-dimensional problems in multiscale mechanics. In particular, the microscale fields and their macroscopic analogues are studied for microstructures that feature material interfaces and material interphases. Analytical solutions are provided to assess the accuracy of the simulation results.
Từ khóa
Tài liệu tham khảo
Matouš K, Geers MGD, Kouznetsova VG, Gillman A (2017) A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys 330:192–220. https://doi.org/10.1016/j.jcp.2016.10.070
Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171(3):387–418. https://doi.org/10.1016/S0045-7825(98)00218-7
Miehe C, Schotte J, Schröder J (1999) Computational micro–macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput Mater Sci 16(1):372–382. https://doi.org/10.1016/S0927-0256(99)00080-4
Feyel F, Chaboche J-L (2000) FE$$^2$$ multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183(3):309–330. https://doi.org/10.1016/S0045-7825(99)00224-8
Kouznetsova VG, Brekelmans WAM, Baaijens FPT (2001) An approach to micro–macro modeling of heterogeneous materials. Comput Mech 27(1):37–48. https://doi.org/10.1007/s004660000212
Ricker S, Mergheim J, Steinmann P, Müller R (2010) A comparison of different approaches in the multi-scale computation of configurational forces. Int J Fracture 166:203–214. https://doi.org/10.1007/s10704-010-9525-2
Özdemir I, Brekelmans WAM, Geers MGD (2008) FE$$^2$$ computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198(3):602–613. https://doi.org/10.1016/j.cma.2008.09.008
Temizer İ, Wriggers P (2011) Homogenization in finite thermoelasticity. J Mech Phys Solids 59(2):344–372. https://doi.org/10.1016/j.jmps.2010.10.004
Sengupta A, Papadopoulos P, Taylor RL (2012) A multiscale finite element method for modeling fully coupled thermomechanical problems in solids. Int J Numer Methods Eng 91(13):1386–1405. https://doi.org/10.1002/nme.4320
Berthelsen R, Denzer R, Oppermann P, Menzel A (2017) Computational homogenisation for thermoviscoplasticity: application to thermally sprayed coatings. Comput Mech 60(5):739–766. https://doi.org/10.1007/s00466-017-1436-x
Berthelsen R, Menzel A (2019) Computational homogenisation of thermo-viscoplastic composites: large strain formulation and weak micro-periodicity. Comput Methods Appl Mech Eng 348:575–603. https://doi.org/10.1016/j.cma.2018.12.032
Schröder J (2009) Derivation of the localization and homogenization conditions for electro-mechanically coupled problems. Comput Mater Sci 46(3):595–599. https://doi.org/10.1016/j.commatsci.2009.03.035
Khalaquzzaman M, Xu B-X, Ricker S, Müller R (2012) Computational homogenization of piezoelectric materials using FE$$^2$$ to determine configurational forces. Tech Mech 32(1):21–37
Keip M-A, Steinmann P, Schröder J (2014) Two-scale computational homogenization of electro-elasticity at finite strains. Comput Methods Appl Mech Eng 278:62–79. https://doi.org/10.1016/j.cma.2014.04.020
Kaiser T, Menzel A (2021) An electro-mechanically coupled computational multiscale formulation for electrical conductors. Arch Appl Mech 91:1509–1526. https://doi.org/10.1007/s00419-020-01837-6
Kaiser T, Menzel A (2021) A finite deformation electro-mechanically coupled computational multiscale formulation for electrical conductors. Acta Mech. https://doi.org/10.1007/s00707-021-03005-5
Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182. https://doi.org/10.1016/j.cam.2009.08.077
Moulinec H, Suquet P (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Comptes rendus de l’Académie des sciences. Série II, Mécanique, physique, chimie, astronomie 318(11):1417–1426
Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157(1):69–94. https://doi.org/10.1016/S0045-7825(97)00218-1
Michel JC, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172(1):109–143. https://doi.org/10.1016/S0045-7825(98)00227-8
Kochmann J, Wulfinghoff S, Ehle L, Mayer J, Svendsen B, Reese S (2018) Efficient and accurate two-scale FE-FFT-based prediction of the effective material behavior of elasto-viscoplastic polycrystals. Comput Mech 61:751–764. https://doi.org/10.1007/s00466-017-1476-2
Frigo M, Johnson SG (2005) The design and implementation of FFTW3. Proc IEEE 93(2):216–231. https://doi.org/10.1109/JPROC.2004.840301
Pekurovsky D (2012) P3DFFT: a framework for parallel computations of Fourier transforms in three dimensions. SIAM J Sci Comput 34(4):192–209. https://doi.org/10.1137/11082748X
Yvonnet J, He Q-C (2007) The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys 233(1):341–368. https://doi.org/10.1016/j.jcp.2006.09.019
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
Barrault M, Maday Y, Nguyen NC, Patera AT (2004) An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. CR Math 339(9):667–672. https://doi.org/10.1016/j.crma.2004.08.006
Hernández JA, Oliver J, Huespe AE, Caicedo MA, Cante JC (2014) High-performance model reduction techniques in computational multiscale homogenization. Comput Methods Appl Mech Eng 276:149–189. https://doi.org/10.1016/j.cma.2014.03.011
Farhat C, Avery P, Chapman T, Cortial J (2014) Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency. Int J Numer Methods Eng 98(9):625–662. https://doi.org/10.1002/nme.4668
Hernández JA, Caicedo MA, Ferrer A (2017) Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Comput Methods Appl Mech Eng 313:687–722. https://doi.org/10.1016/j.cma.2016.10.022
van Tuijl RA, Remmers JJC, Geers MGD (2018) Integration efficiency for model reduction in micro-mechanical analyses. Comput Mech 62:151–169. https://doi.org/10.1007/s00466-017-1490-4
Li B, Chen X (2014) Wavelet-based numerical analysis: a review and classification. Finite Elem Anal Des 81:14–31. https://doi.org/10.1016/j.finel.2013.11.001
Morlet J, Arens G, Fourgeau E, Glard D (1982) Wave propagation and sampling theory—part I: complex signal and scattering in multilayered media. Geophysics 47(4):203–221. https://doi.org/10.1190/1.1441328
Goedecker S (1998) Wavelets and their application for the solution of partial differential equations in physics. Presses Polytechniques et Universitaires Romandes, Lausanne
Goedecker S (2009) Wavelets and their application for the solution of Poisson’s and Schrödinger’s equation. In: Grotendorst J, Attig N, Blügel S, Marx D (eds) Multiscale simulation methods in molecular sciences. Institute for Advanced Simulation, Forschungszentrum Jülich, pp 507–534
Vasilyev OV, Bowman C (2000) Second-generation wavelet collocation method for the solution of partial differential equations. J Comput Phys 165:660–693. https://doi.org/10.1006/jcph.2000.6638
Harnish C, Matouš K, Livescu D (2018) Adaptive wavelet algorithm for solving nonlinear initial-boundary value problems with error control. Int J Multiscale Comput Eng 16(1):19–43. https://doi.org/10.1615/IntJMultCompEng.2018024915
Harnish C, Dalessandro L, Matouš K, Livescu D (2021) A multiresolution adaptive wavelet method for nonlinear partial differential equations. Int J Multiscale Comput Eng 19(2):29–37. https://doi.org/10.1615/IntJMultCompEng.2021039451
Amaratunga K, Williams JR (1993) Wavelet based Greens’s function approach to 2D PDEs. Eng Comput 10:349–367. https://doi.org/10.1108/eb023913
Amaratunga K, Williams JR, Qian S, Weiss J (1994) Wavelet-Galerkin solutions for one-dimensional partial differential equations. Int J Numer Methods Eng 37(16):2703–2716. https://doi.org/10.1002/nme.1620371602
Jones S, Legrand M (2012) The wavelet-Galerkin method for solving PDE’s with spatially de-pendent variables. In: 19th International Congress on Sound and Vibration (ICSV19), pp 1–8
Christon MA, Roach DW (2000) The numerical performance of wavelets for PDEs: the multi-scale finite element. Comput Mech 25:230–244. https://doi.org/10.1007/s004660050472
Naldi G, Venini P (1997) Wavelet analysis of structures: statics, dynamics and damage identification. Meccanica 32:223–230. https://doi.org/10.1007/s11012-004-3200-5
Kim JE, Jang G-W, Kim YY (2003) Adaptive multiscale wavelet-Galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization. Int J Solids Struct 40(23):6473–6496. https://doi.org/10.1016/S0020-7683(03)00417-7
Azdoud Y, Ghosh S (2017) Adaptive wavelet-enriched hierarchical finite element model for polycrystalline microstructures. Comput Methods Appl Mech Eng 370:337–360. https://doi.org/10.1016/j.cma.2017.04.018
Azdoud Y, Cheng J, Ghosh S (2017) Wavelet-enriched adaptive crystal plasticity finite element model for polycrystalline microstructures. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2017.08.026
van Tuijl RA, Harnish C, Matouš K, Remmers JJC, Geers MGD (2019) Wavelet based reduced order models for microstructural analyses. Comput Mech 63(3):535–554. https://doi.org/10.1007/s00466-018-1608-3
van Tuijl RA, Remmers JJC, Geers MGD (2020) Multi-dimensional wavelet reduction for the homogenisation of microstructures. Comput Methods Appl Mech Eng 359:112652. https://doi.org/10.1016/j.cma.2019.112652
Daubechies I (1992) Ten lectures on wavelets. Society for Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9781611970104
Williams JR, Amaratunga K (1994) Introduction to wavelets in engineering. Int J Numer Methods Eng 37(14):2365–2388. https://doi.org/10.1002/nme.1620371403
Sweldens W (1996) The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl Comput Harmon Anal 3(2):186–200. https://doi.org/10.1006/acha.1996.0015
Sweldens W (1998) The lifting scheme: a construction of second generation wavelets. SIAM J Math Anal 29(2):511–546. https://doi.org/10.1137/S0036141095289051
Rioul O (1992) Simple regularity criteria for subdivision schemes. SIAM J Math Anal 23(6):1544–1576. https://doi.org/10.1137/0523086