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Consistent discretization of higher-order interface models for thin layers and elastic material surfaces, enabled by isogeometric cut-cell methods. (English) Zbl 1441.74031
Summary: Many interface formulations, e.g. based on asymptotic thin interphase models or material surface theories, involve higher-order differential operators and discontinuous solution fields. In this article, we are taking first steps towards a variationally consistent discretization framework that naturally accommodates these two challenges by synergistically combining recent developments in isogeometric analysis and cut-cell finite element methods. Its basis is the mixed variational formulation of the elastic interface problem that provides access to jumps in displacements and stresses for incorporating general interface conditions. Upon discretization with smooth splines, derivatives of arbitrary order can be consistently evaluated, while cut-cell meshes enable discontinuous solutions at potentially complex interfaces. We demonstrate via numerical tests for three specific nontrivial interfaces (two regimes of the Benveniste-Miloh classification of thin layers and the Gurtin-Murdoch material surface model) that our framework is geometrically flexible and provides optimal higher-order accuracy in the bulk and at the interface.

MSC:
 74A50 Structured surfaces and interfaces, coexistent phases 65D07 Numerical computation using splines 74S05 Finite element methods applied to problems in solid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
CutFEM
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 [1] Achenbach, J. D.; Zhu, H., Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites, J. Mech. Phys. Solids, 37, 381-393 (1989) [2] Gibson, R. F., Principles of Composite Material Mechanics (2011), CRC press [3] Chhapadia, P.; Mohammadi, P.; Sharma, P., Curvature-dependent surface energy and implications for nanostructures, J. Mech. Phys. Solids, 59, 2103-2115 (2011) · Zbl 1270.74018 [4] Lee, H.; Dellatore, S. M.; Miller, V. M.; Messersmith, P. B., Mussel-inspired surface chemistry for multifunctional coatings, Science, 318, 426-430 (2007) [5] Karger-Kocsis, J.; Mahmood, H.; Pegoretti, A., Recent advances in fiber/matrix interphase engineering for polymer composites, Prog. Mater. Sci., 1-43 (2015) [6] Gao, S.-L.; Mäder, E., Characterisation of interphase nanoscale property variations in glass fibre reinforced polypropylene and epoxy resin composites, Composites A, 33, 559-576 (2002) [7] Jancar, J., Review of the role of the interphase in the control of composite performance on micro-and nano-length scales, J. Mater. Sci., 43, 6747-6757 (2008) [8] Givoli, D., Finite element modeling of thin layers, Comput. Model. Eng. Sci., 5, 497-514 (2004) [9] Eslami, H.; Müller-Plathe, F., How thick is the interphase in an ultrathin polymer film? coarse-grained molecular dynamics simulations of polyamide-6,6 on graphene, J. Phys. Chem. C, 117, 5249-5257 (2013) [10] Wang, Z.; Lv, Q.; Chen, S.; Li, C.; Sun, S.; Hu, S., Effect of interfacial bonding on interphase properties in SiO2/epoxy nanocomposite: a molecular dynamics simulation study, ACS Appl. Mater. Interfaces, 8, 7499-7508 (2016) [11] Bövik, P., On the modelling of thin interface layers in elastic and acoustic scattering problems, Q. J. Mech. Appl. Math., 47, 17-42 (1994) · Zbl 0803.73025 [12] Klarbring, A.; Movchan, A. B., Asymptotic modelling of adhesive joints, Mech. Mater., 28, 137-145 (1998) [13] Bigoni, D.; Serkov, S. K.; Valentini, M.; Movchan, A. B., Asymptotic models of dilute composites with imperfectly bonded inclusions, Int. J. Solids Struct., 35, 3239-3258 (1998) · Zbl 0918.73042 [14] Hashin, Z., Thin interphase/imperfect interface in elasticity with application to coated fiber composites, J. Mech. Phys. Solids, 50, 2509-2537 (2002) · Zbl 1080.74006 [15] Gu, S. T.; He, Q. C., Interfacial discontinuity relations for coupled multifield phenomena and their application to the modeling of thin interphases as imperfect interfaces, J. Mech. Phys. Solids, 59, 1413-1426 (2011) · Zbl 1270.74037 [16] Benveniste, Y., The effective mechanical behaviour of composite materials with imperfect contact between the constituents, Mech. Mater., 4, 197-208 (1985) [17] Hashin, Z., Thermoelastic properties of fiber composites with imperfect interface, Mech. Mater., 8, 333-348 (1990) [18] Hashin, Z., The spherical inclusion with imperfect interface, J. Appl. Mech., 58, 444-449 (1991) [19] Bigoni, D.; Movchan, A. B., Statics and dynamics of structural interfaces in elasticity, Int. J. Solids Struct., 39, 4843-4865 (2002) · Zbl 1042.74034 [20] Duan, H. L.; Wang, J.; Huang, Z. P.; Luo, Z. Y., Stress concentration tensors of inhomogeneities with interface effects, Mech. Mater., 37, 723-736 (2005) [21] Wang, J.; Duan, H. L.; Zhang, Z.; Huang, Z. P., An anti-interpenetration model and connections between interphase and interface models in particle-reinforced composites, Int. J. Mech. Sci., 47, 701-708 (2005) · Zbl 1192.74084 [22] Benveniste, Y.; Miloh, T., Imperfect soft and stiff interfaces in two-dimensional elasticity, Mech. Mater., 33, 309-323 (2001) [23] Gurtin, M. E.; Murdoch, A. I., A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 291-323 (1975) · Zbl 0326.73001 [24] Gurtin, M. E.; Murdoch, A. I., Surface stress in solids, Int. J. Solids Struct., 14, 431-440 (1978) · Zbl 0377.73001 [25] Gurtin, M. E.; Weissmüller, J.; Larché, F., A general theory of curved deformable interfaces in solids at equilibrium, Phil. Mag. A, 78, 1093-1109 (1998) [26] Steigmann, D. J.; Ogden, R. W., Plain deformations of elastic solids with intrinsic boundary elasticity, Proc. R. Soc. Lond. A, 453, 853-877 (1997) · Zbl 0938.74014 [27] Steigmann, D. J.; Ogden, R. W., Elastic surface-substrate interactions, Proc. R. Soc. Lond. A, 455, 437-474 (1999) · Zbl 0926.74016 [28] Javili, A.; Dell’ Isola, F.; Steinmann, P., Geometrically nonlinear higher-gradient elasticity with energetic boundaries, J. Mech. Phys. Solids, 61, 2381-2401 (2013) · Zbl 1294.74014 [29] Javili, A.; McBride, A.; Steinmann, P.; Reddy, B. D., A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology, Comput. Mech., 54, 745-762 (2014) · Zbl 1311.74091 [30] Duan, H. L.; Karihaloo, B. L., Thermo-elastic properties of heterogeneous materials with imperfect interfaces: Generalized levin’s formula and hill’s connections, J. Mech. Phys. Solids, 55, 1036-1052 (2007) · Zbl 1170.74017 [31] Mogilevskaya, S. G.; Crouch, S. L.; Stolarski, H. K., Multiple interacting circular nano-inhomogeneities with surface/interface effects, J. Mech. Phys. Solids, 56, 2298-2327 (2008) · Zbl 1171.74398 [32] Zemlyanova, A. Y.; Mogilevskaya, S. G., Circular inhomogeneity with Steigmann-Ogden interface: Local fields, neutrality, and Maxwell’s type approximation formula, Int. J. Solids Struct., 135, 85-98 (2018) [33] Han, Z.; Mogilevskaya, S. G.; Schillinger, D., Local fields and overall transverse properties of unidirectional composite materials with multiple nanofibers and Steigmann-Ogden interfaces, Int. J. Solids Struct., 147, 166-182 (2018) [34] Capdeville, Y.; Marigo, J. J., Shallow layer correction for spectral element like methods, Geophys. J. Int., 172, 1135-1150 (2008) [35] Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1973), Prentice-Hall · Zbl 0278.65116 [36] Sussmann, C.; Givoli, D.; Benveniste, Y., Combined asymptotic finite-element modeling of thin layers for scalar elliptic problems, Comput. Methods Appl. Mech. Engrg., 200, 3265-3269 (2011) · Zbl 1230.80011 [37] Dumont, S.; Lebon, F.; Rizzoni, R., An asymptotic approach to the adhesion of thin stiff films, Mech. Res. Commun., 58, 24-35 (2014) [38] Yvonnet, J.; Quang, H. Le; He, Q.-C., An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites, Comput. Mech., 42, 1, 119-131 (2008) · Zbl 1188.74076 [39] Zhu, Q.-Z.; Gu, S.-T.; Yvonnet, J.; Shao, J.-F.; He, Q.-C., Three-dimensional numerical modelling by XFEM of spring-layer imperfect curved interfaces with applications to linearly elastic composite materials, Internat. J. Numer. Methods Engrg., 88, 4, 307-328 (2011) · Zbl 1242.74181 [40] Benvenuti, E.; Ventura, G.; Ponara, N.; Tralli, A., Variationally consistent eXtended FE model for 3D planar and curved imperfect interfaces, Comput. Methods Appl. Mech. Engrg., 267, 434-457 (2013) · Zbl 1286.74095 [41] Javili, A.; F. Dell’ Isola, F.; Steinmann, P., Geometrically nonlinear higher-gradient elasticity with energetic boundaries, J. Mech. Phys. Solids, 61, 2381-2401 (2013) · Zbl 1294.74014 [42] Javili, A.; McBride, A.; Steinmann, P.; Reddy, B. D., A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology, Comput. Mech., 54, 745-762 (2014) · Zbl 1311.74091 [43] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 4135-4195 (2005) · Zbl 1151.74419 [44] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric analysis: Towards Integration of CAD and FEA (2009), John Wiley & Sons · Zbl 1378.65009 [45] Schillinger, D.; Ruess, M., 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 (2015) · Zbl 1348.65056 [46] Burman, E.; Claus, S.; Hansbo, P.; Larson, M.-G.; Massing, A., CutFEM: discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., 104, 7, 472-501 (2015) · Zbl 1352.65604 [47] Hansbo, A.; Hansbo, P., An unfitted finite element method, based on nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191, 537-552 (2002) · Zbl 1035.65125 [48] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, D. L., Unified analysis of discontinuous galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2002) · Zbl 1008.65080 [49] Benveniste, Y., Exact results for the local fields and the effective moduli of fibrous composites with thickly coated fibers, J. Mech. Phys. Solids, 71, 219-238 (2014) · Zbl 1328.74032 [50] Schillinger, D.; Ruess, M.; Zander, N.; Bazilevs, Y.; Düster, A.; Rank, E., Small and large deformation analysis with the $$p$$- and B-spline versions of the Finite Cell Method, Comput. Mech., 50(4), 445-478 (2012) · Zbl 1398.74401 [51] Schillinger, D.; Dede’, L.; Scott, M. A.; Evans, J. A.; Borden, M. J.; Rank, E.; Hughes, T. J.R., An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Comput. Methods Appl. Mech. Engrg., 249-250, 116-150 (2012) · Zbl 1348.65055 [52] Gomez, H.; Calo, V. M.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of the Cahn-Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., 197, 4333-4352 (2008) · Zbl 1194.74524 [53] Verhoosel, C. V.; Scott, M. A.; Hughes, T. J.R.; De Borst, R., An isogeometric analysis approach to gradient damage models, Internat. J. Numer. Methods Engrg., 86, 1, 115-134 (2011) · Zbl 1235.74320 [54] Kiendl, J.; Auricchio, F.; Hughes, T. J.R.; Reali, A., Single-variable formulations and isogeometric discretizations for shear deformable beams, Comput. Methods Appl. Mech. Engrg., 284, 988-1004 (2015) · Zbl 1423.74492 [55] Zhao, Y.; Schillinger, D.; Xu, B.-X., Variational boundary conditions based on the Nitsche method for fitted and unfitted isogeometric discretizations of the mechanically coupled Cahn-Hilliard equation, J. Comput. Phys., 340, 177-199 (2017) · Zbl 1380.65293 [56] Piegl, L.; Tiller, W., The NURBS Book (1997), Springer · Zbl 0868.68106 [57] Martin, T.; Cohen, E.; Kirby, R. M., Volumetric parameterization and trivariate b-spline fitting using harmonic functions, Comput. Aided Geom. Design, 26, 6, 648-664 (2009) · Zbl 1205.65094 [58] Cohen, E.; Martin, T.; Kirby, R. M.; Lyche, T.; Riesenfeld, R. F., Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 199, 334-356 (2010) · Zbl 1227.74109 [59] Haberleitner, M.; Jttler, B.; Scott, M. A.; Thomas, D. C., Isogeometric analysis: Representation of geometry, (Encyclopedia of Computational Mechanics Second Edition (2018), John Wiley & Sons, Ltd) [60] Müller, B.; Kummer, F.; Oberlack, M., Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Internat. J. Numer. Methods Engrg., 96, 8, 512-528 (2013) · Zbl 1352.65083 [61] Fries, T.-P.; Omerovic, S., Higher-order accurate integration of implicit geometries, Internat. J. Numer. Methods Engrg., 106, 1, 323-371 (2016) · Zbl 1352.65498 [62] Kudela, L.; Zander, N.; Bog, T.; Kollmannsberger, S.; Rank, E., Efficient and accurate numerical quadrature for immersed boundary methods, Adv. Model. Simul. Eng. Sci., 2, 1, 1-22 (2015) [63] Stavrev, A.; Nguyen, L. H.; Shen, R.; Varduhn, V.; Behr, M.; Elgeti, S.; Schillinger, D., Geometrically accurate, efficient, and flexible quadrature techniques for the tetrahedral finite cell method, Comput. Methods Appl. Mech. Engrg., 310, 646-673 (2016) [64] Lehrenfeld, C., High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Engrg., 300, 716-733 (2016) · Zbl 1425.65168 [65] Embar, A.; Dolbow, J.; Harari, I., Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements, Internat. J. Numer. Methods Engrg., 83, 877-898 (2010) · Zbl 1197.74178 [66] Burman, E.; Hansbo, P., Fictitious domain finite element methods using cut elements: Ii. a stabilized Nitsche method, Appl. Numer. Math., 62, 4, 328-341 (2012) · Zbl 1316.65099 [67] Jiang, W.; Annavarapu, C.; Dolbow, J. E.; Harari, I., A robust Nitsche’s formulation for interface problems with spline-based finite elements, Internat. J. Numer. Methods Engrg., 104, 7, 676-696 (2015) · Zbl 1352.65515 [68] Schillinger, D.; Harari, I.; Hsu, M.-C.; Kamensky, D.; Stoter, K. F.S.; Yu, Y.; Zhao, Y., The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements, Comput. Methods Appl. Mech. Engrg., 309, 625-652 (2016) [70] Massing, A.; Schott, B.; Wall, W. A., A stabilized Nitsche cut finite element method for the Oseen problem, Comput. Methods Appl. Mech. Engrg. (2017) [71] Düster, A.; Parvizian, J.; Yang, Z.; Rank, E., The finite cell method for three-dimensional problems of solid mechanics, Comput. Methods Appl. Mech. Engrg., 197, 3768-3782 (2008) · Zbl 1194.74517 [72] Joulaian, M.; Hubrich, S.; Düster, A., Numerical integration of discontinuities on arbitrary domains based on moment fitting, Comput. Mech., 57, 6, 979-999 (2016) · Zbl 1382.65066 [73] Bouclier, R.; Passieux, J.-C., A Nitsche-based non-intrusive coupling strategy for global/local isogeometric structural analysis, Comput. Methods Appl. Mech. Engrg., 340, 253-277 (2018) [74] Rivière, B., Linear elasticity, (Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations (2008), Society for Industrial and Applied Mathematics), 109-115, chapter 5 [75] Soon, S.-C.; Cockburn, B.; Stolarski, H. K., A hybridizable discontinuous Galerkin method for linear elasticity, Internat. J. Numer. Methods Engrg., 80, 8, 1058-1092 (2009) · Zbl 1176.74196 [76] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer · Zbl 0788.73002 [77] Hansbo, P.; Larson, M. G., Discontinuous galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comput. Methods Appl. Mech. Engrg., 191, 17, 1895-1908 (2002) · Zbl 1098.74693 [78] Annavarapu, C.; Hautefeuille, M.; Dolbow, J. E., A robust Nitsche’s formulation for interface problems, Comput. Methods Appl. Mech. Engrg., 225, 44-54 (2012) · Zbl 1253.74096 [79] Harari, I.; Grosu, E., A unified approach for embedded boundary conditions for fourth-order elliptic problems, Internat. J. Numer. Methods Engrg., 104, 7, 655-675 (2015) · Zbl 1352.65507 [80] Langer, U.; Mantzaflaris, A.; Moore, S.; Toulopoulos, I., Multipatch discontinuous Galerkin isogeometric analysis, (Isogeometric Analysis and Applications. Isogeometric Analysis and Applications, Lecture Notes in Computational Science and Engineering (2015), Springer), 1-32 · Zbl 1334.65194 [81] Miller, R. E.; Shenoy, V. B., Size-dependent elastic properties of nanosized structural elements, Nanotechnology, 11, 3, 139 (2000) [82] Sharma, P.; Ganti, S., Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies, J. Appl. Mech., 71, 5, 663-671 (2004) · Zbl 1111.74629 [83] Javili, A.; Steinmann, P.; Mosler, J., Micro-to-macro transition accounting for general imperfect interfaces, Comput. Methods Appl. Mech. Engrg., 317, 274-317 (2017) [84] Cenanovic, M.; Hansbo, P.; Larson, M. G., Cut finite element modeling of linear membranes, Comput. Methods Appl. Mech. Engrg., 310, 98-111 (2016) [85] Schillinger, D.; Gangwar, T.; Gilmanov, A.; Heuschele, J. D.; Stolarski, H. K., Embedded shell finite elements: Solid – shell interaction, surface locking, and application to image-based bio-structures, Comput. Methods Appl. Mech. Engrg., 335, 298-326 (2018) [86] Guo, Y.; Heller, J.; Hughes, T. J.R.; Ruess, M.; Schillinger, D., Variationally consistent isogeometric analysis of trimmed thin shells at finite deformations, based on the step exchange format, Comput. Methods Appl. Mech. Engrg., 336, 39-79 (2018)
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