Robust modelling of implicit interfaces by the scaled boundary finite element method. (English) Zbl 1464.74261
Summary: In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is \(\mathcal{O}(h^2)\) in \(L^2\) norm and \(\mathcal{O}(h)\) in the \(H^1\) semi-norm. Further the condition number increases with the mesh size similar to the FEM.
MSC:
| 74S15 | Boundary element methods applied to problems in solid mechanics |
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 74A50 | Structured surfaces and interfaces, coexistent phases |
Keywords:
bi-material interfaces; Dirichlet boundary conditions on interfaces; material inclusions; scaled boundary finite element method; weak discontinuityReferences:
| [1] | Shahil, K. M.; Balandin, A. A., Graphene-multilayer graphene nanocomposites as highly efficient thermal interface materials, Nano Lett, 12, 2, 861-867 (2012) |
| [2] | Deresiewicz, H.; Rice, J., The effect of boundaries on wave propagation in a liquid-filled porous solid: V. transmission across a plane interface, Bull Seismol Soc Am, 54, 1, 409-416 (1964) |
| [3] | Komatitsch, D.; Barnes, C.; Tromp, J., Wave propagation near a fluid-solid interface: a spectral-element approach, Geophysics, 65, 2, 623-631 (2000) |
| [4] | Sarabandi, K.; Koh, I.-S., Effect of canopy-air interface roughness on HF-VHF wave propagation in forest, IEEE Trans Antennas Propag, 50, 2, 111-121 (2002) |
| [5] | Singamaneni, S.; LeMieux, M. C.; Lang, H. P.; Gerber, C.; Lam, Y.; Zauscher, S., Bimaterial microcantilevers as a hybrid sensing platform, Adv Mater, 20, 4, 653-680 (2008) |
| [6] | Rochus, V.; Van Miegroet, L.; Rixen, D. J.; Duysinx, P., Electrostatic simulation using xfem for conductor and dielectric interfaces, Int J Numer Methods Eng, 85, 10, 1207-1226 (2011) · Zbl 1217.78061 |
| [7] | Kattis, M.; Mavroyannis, G., Feeble interfaces in bimaterials, Acta Mech, 185, 1-2, 11-29 (2006) · Zbl 1114.74014 |
| [8] | Narayanaswamy, A.; Gu, N., Heat transfer from freely suspended bimaterial microcantilevers, J Heat Transfer, 133, 4 (2011) |
| [9] | Hansbo, P., Nitsche’S method for interface problems in computa-tional mechanics, GAMM-Mitteilungen, 28, 2, 183-206 (2005) · Zbl 1179.65147 |
| [10] | Yang, Z.; Deeks, A., Calculation of transient dynamic stress intensity factors at bimaterial interface cracks using a sbfem-based frequency-domain approach, Sci China Ser G Phys Mech Astronomy, 51, 5, 519-531 (2008) |
| [11] | Lin, T.; Sheen, D.; Zhang, X., A locking-free immersed finite element method for planar elasticity interface problems, J Comput Phys, 247, 228-247 (2013) · Zbl 1349.74328 |
| [12] | Li, C.; Song, C.; Man, H.; Ooi, E. T.; Gao, W., 2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the sbfem, Int J Solids Struct, 51, 11-12, 2096-2108 (2014) |
| [13] | Barrett, J. W.; Elliott, C. M., Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J Numer Anal, 7, 3, 283-300 (1987) · Zbl 0629.65118 |
| [14] | Leveque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J Numer Anal, 31, 4, 1019-1044 (1994) · Zbl 0811.65083 |
| [15] | Li, Z.; Ito, K., The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains, 33 (2006), SIAM · Zbl 1122.65096 |
| [16] | Zhou, Y.; Zhao, S.; Feig, M.; Wei, G., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J Comput Phys, 213, 1, 1-30 (2006) · Zbl 1089.65117 |
| [17] | Zhao, S.; Wei, G., High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces, J Comput Phys, 200, 1, 60-103 (2004) · Zbl 1050.78018 |
| [18] | Wang, S.; Nan, C.; Qiao, J.; Huang, D.; Nabipour, N.; Ross, D., Free convection and entropy generation in a nanofluid-filled star-ellipse annulus using lattice boltzmann method supported by immersed boundary method, Int J Mech Sci, 105526 (2020) |
| [19] | Fries, T.-P.; Belytschko, T., The extended/generalized finite element method: an overview of the method and its applications, Int J Numer Methods Eng, 84, 3, 253-304 (2010) · Zbl 1202.74169 |
| [20] | Bhattacharya, S.; Singh, I.; Mishra, B., Fatigue life simulation of functionally graded materials under cyclic thermal load using xfem, Int J Mech Sci, 82, 41-59 (2014) |
| [21] | Stolarska, M.; Chopp, D. L.; Moës, N.; Belytschko, T., Modelling crack growth by level sets in the extended finite element method, Int J Numer Methods Eng, 51, 8, 943-960 (2001) · Zbl 1022.74049 |
| [22] | Bouhala, L.; Makradi, A.; Belouettar, S., Thermo-anisotropic crack propagation by xfem, Int J Mech Sci, 103, 235-246 (2015) |
| [23] | Sukumar, N.; Chopp, D.; Moës, N.; Belytschko, T., Modeling holes and inclusions by level sets in the extended finite-element method, Comput Methods Appl Mech Eng, 190, 46, 6183-6200 (2001) · Zbl 1029.74049 |
| [24] | Patil, R.; Mishra, B.; Singh, I., A new multiscale xfem for the elastic properties evaluation of heterogeneous materials, Int J Mech Sci, 122, 277-287 (2017) |
| [25] | Motamedi, D.; Mohammadi, S., Fracture analysis of composites by time independent moving-crack orthotropic xfem, Int J Mech Sci, 54, 1, 20-37 (2012) |
| [26] | Burman, E.; Claus, S.; Hanbso, P.; Larson, M.; Massing, A., CutFEM: discretizing geometry and partial differential equations, Int J Numer Methods Eng, 104, 472-501 (2015) · Zbl 1352.65604 |
| [27] | Hansbo, P.; Larson, M. G.; Larsson, K., Cut finite element methods for linear elasticity problems, (Bordas, S.; Burman, E.; Larson, M.; Olshanskii, M., Geometrically Unfitted Finite Element Methods and Applications. Geometrically Unfitted Finite Element Methods and Applications, Lecture Notes in Computational Science and Engineering, 121 (2017), Springer International), 25-63 · Zbl 1390.74180 |
| [28] | Lozinski, A., CutFEM without cutting the mesh cells: a new way to impose Dirichlet and Neumann boundary conditions on unfitted meshes, Comput Methods Appl Mech Eng, 356, 75-100 (2019) · Zbl 1441.65108 |
| [29] | Ji, H.; Dolbow, J., On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method, Int J Numer Methods Eng, 61, 14, 2508-2535 (2004) · Zbl 1075.74651 |
| [30] | Burman, E.; Hansbo, P., Fictitious domain finite element methods using cut elements: i. a stabilized lagrange multiplier method, Comput Methods Appl Mech Eng, 199, 41-44, 2680-2686 (2010) · Zbl 1231.65207 |
| [31] | Claus, S.; Kerfriden, P., A cutfem method for two-phase flow problems, Comput Methods Appl Mech Eng, 348, 185-206 (2019) · Zbl 1440.76054 |
| [32] | Burman, E.; Hansbo, P.; Larson, M. G.; Zahedi, S., Cut finite element methods for coupled bulk-surface problems, Numerische Mathematik, 133, 2, 203-231 (2016) · Zbl 1341.65044 |
| [33] | Fries, T.-P.; Byfut, A.; Alizada, A.; Cheng, K. W.; Schröder, A., Hanging nodes and XFEM, Int J Numer Methods Eng, 86, 404-430 (2010) · Zbl 1216.74020 |
| [34] | Joulaian, M.; Düster, A., Local enrichment of the finite cell method for problems with material interfaces, Comput Mech, 52, 741-762 (2013) · Zbl 1311.74123 |
| [35] | Teng, Z.; Liao, D.; Wu, S.; Sun, F.; Chen, T.; Zhang, Z., An adaptively refined XFEM for the dynamic fracture problems with micro-defects, Theor Appl Fract Mech, 103, 102255 (2019) |
| [36] | Song, C.; Wolf, J. P., The scaled boundary finite-element methodalias consistent infinitesimal finite-element cell methodfor elastodynamics, Comput Methods Appl Mech Eng, 147, 3-4, 329-355 (1997) · Zbl 0897.73069 |
| [37] | Yang, Z., Fully automatic modelling of mixed-mode crack propagation using scaled boundary finite element method, Eng Fract Mech, 73, 12, 1711-1731 (2006) |
| [38] | Song, C.; Wolf, J. P., Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method, Comput Struct, 80, 183-197 (2002) |
| [39] | Bird, G.; Trevelyan, J.; Augarde, C., A coupled BEM/scaled boundary FEM formulation for accurate computations in linear elastic fracture mechanics, Eng Anal Bound Elem, 34, 599-610 (2010) · Zbl 1267.74120 |
| [40] | Garg, N.; Chakladar, N. D.; Prusty, B. G.; Song, C.; Phillips, A. W., Modelling of laminated composite plates with weakly bonded interfaces using scaled boundary finite element method, Int J Mech Sci, 170, 105349 (2020) |
| [41] | Wang, W.; Ye, W.; Ren, L.; Jiang, Y., A scaled boundary finite element method for bending analysis of fiber-reinforced piezoelectric laminated composite plates, Int J Mech Sci, 161, 105011 (2019) |
| [42] | Eisenträger, J.; Zhang, J.; Song, C.; Eisenträger, S., An sbfem approach for rate-dependent inelasticity with application to image-based analysis, Int J Mech Sci, 105778 (2020) |
| [43] | Song, C.; Wolf, J., Consistent infinitesimal finite-element cell method: threedimensional vector wave equation, Int J Numer Methods Eng, 39, 2189-2208 (1996) · Zbl 0885.73089 |
| [44] | Gravenkamp, H.; Saputra, A. A.; Song, C.; Birk, C., Efficient wave propagation simulation on quadtree meshes using SBFEM with reduced modal basis, Int J Numer Methods Eng, 110, 1119-1141 (2017) · Zbl 07866624 |
| [45] | Liu, L.; Zhang, J.; Song, C.; Birk, C.; Gao, W., An automatic approach for the acoustic analysis of three-dimensional bounded and unbounded domains by scaled boundary finite element method, Int J Mech Sci, 151, 563-581 (2019) |
| [46] | Liu, Y.; Saputra, A. A.; Wang, J.; Tin-Loi, F.; Song, C., Automatic polyhedral mesh generation and scaled boundary finite element analysis of stl models, Comput Methods Appl Mech Eng, 313, 106-132 (2017) · Zbl 1439.74444 |
| [47] | Natarajan, S.; Song, C., Representation of singular fields without asymptotic enrichment in the extended finite element method, Int J Numer Methods Eng, 96, 13, 813-841 (2013) · Zbl 1352.74298 |
| [48] | Natarajan, S.; Song, C.; Belouettar, S., Numerical evaluation of stress intensity factors and t-stress for interfacial cracks and cracks terminating at the interface without asymptotic enrichment, Comput Methods Appl Mech Eng, 279, 86-112 (2014) · Zbl 1423.74364 |
| [49] | Song, C.; Ooi, E. T.; Natarajan, S., A review of the scaled boundary finite element method for two-dimensional linear elastic fracture mechanics, Eng Fract Mech, 187, 45-73 (2018) |
| [50] | Song, C.; Wolf, J. P., The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics, Comput Methods Appl Mech Eng, 147, 3-4, 329-355 (1997) · Zbl 0897.73069 |
| [51] | Song, C., A matrix function solution for the scaled boundary finite element equation in statics., Comput Methods Appl Mech Eng, 193, 2325-2356 (2004) · Zbl 1067.74586 |
| [52] | Ooi, E. T.; Song, C.; Natarajan, S., A scaled boundary finite element formulation for poroelasticity, Int J Numer Methods Eng, 114, 905-929 (2018) · Zbl 07878389 |
| [53] | Moumnassi, M.; Belouettar, S.; Béchet, E.; Bordas, S. P., Finite element analysis on implicitly defined domains: an accurate representation based on arbitrary parametric surfaces, Comput Methods Appl Mech Eng, 200, 774-796 (2011) · Zbl 1225.65111 |
| [54] | Moumnassi, M.; Bordas, S.; Figueredo, R.; Sansen, P., Analysis using higher-order XFEM: implicit representation of geometrical features from a given parametric representation, Mech Ind, 15, 443-448 (2014) |
| [55] | Natarajan, S.; Ooi, E. T.; Song, C., Finite element computations over quadtree meshes: strain smoothing and semi-analytical formulation, Int J Adv Eng Sci Appl Math, 7, 124-133 (2015) · Zbl 1342.74168 |
| [56] | Hansbo, A.; Hansbo, P., An unfitted finite element method, based on nitsches method, for elliptic interface problems, Comput Methods Appl Mech Eng, 191, 47-48, 5537-5552 (2002) · Zbl 1035.65125 |
| [57] | Vaughan, B.; Smith, B.; Chopp, D., A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources, Comm App Math Comp Sci, 1, 1, 207-228 (2007) · Zbl 1153.65373 |
| [58] | Wolf, J. P., The scaled boundary finite element method (2003), John Wiley & Sons |
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