Abstract
Virtual element methods (VEM) provide great flexibility in solving numerical problems defined on arbitrarily shaped polygonal or polyhedral discretizations. In this paper, we develop a framework for two dimensional elastic problems defined on complex topology models using high-order virtual element methods from an engineering perspective. The VEM discrete formulations are detailedly derived following the rule used in standard FEM. An arbitrarily complex model is first embedded into a rectangular domain which is then discretized into a structured grid. The elements intersecting with the boundaries are further adaptively refined through a quad-tree refinement strategy controlled by a subdivision level or an approximation error. An optimization method is proposed to avoid the generation of tiny elements and two averaged schemes for stress recovery in post-processing are discussed. The behavior of the proposed VEM is thoroughly studied and the results are compared with analytical solutions and that obtained from FEM. The heavy burden placed on meshing complex CAD geometries is greatly alleviated and the convergence studies confirm the accuracy and convergence of the method.
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References
Ahmad B, Alsaedi A, Brezzi F, Marini LD, Russo A (2013) Equivalent projectors for virtual element methods. Comput Math Appl 66(3):376–391
Aldakheel F, Hudobivnik B, Artioli E, da Veiga LB, Wriggers P (2020) Curvilinear virtual elements for contact mechanics. Comput Methods Appl Mech Eng 372:113394
Aldakheel F, Hudobivnik B, Hussein A, Wriggers P (2018) Phase-field modeling of brittle fracture using an efficient virtual element scheme. Comput Methods Appl Mech Eng 341:443–466
Aldakheel F, Hudobivnik B, Wriggers P (2019) Virtual element formulation for phase-field modeling of ductile fracture. Int J Multiscale Comput Eng 17(2):181
Aldakheel F, Hudobivnik B, Wriggers P (2019) Virtual elements for finite thermo-plasticity problems. Comput Mech 64(5):1347–1360
Antonietti PF, Bruggi M, Scacchi S, Verani M (2017) On the virtual element method for topology optimization on polygonal meshes: a numerical study. Comput Math Appl 74(5):1091–1109
Aragón AM, Liang B, Ahmadian H, Soghrati S (2020) On the stability and interpolating properties of the hierarchical interface-enriched finite element method. Comput Methods Appl Mech Eng 362:112671
Artioli E, Da Veiga LB, Dassi F (2020) Curvilinear virtual elements for 2D solid mechanics applications. Comput Methods Appl Mech Eng 359:112667
Artioli E, Da Veiga LB, Lovadina C, Sacco E (2017) Arbitrary order 2D virtual elements for polygonal meshes: part I, elastic problem. Comput Mech 60(3):355–377
Artioli E, De Miranda S, Lovadina C, Patruno L (2017) A stress/displacement virtual element method for plane elasticity problems. Comput Methods Appl Mech Eng 325:155–174
Artioli E, de Miranda S, Lovadina C, Patruno L (2019) An equilibrium-based stress recovery procedure for the VEM. Int J Numer Methods Eng 117(8):885–900
Artioli E, da Veiga LB, Verani M (2020) An adaptive curved virtual element method for the statistical homogenization of random fibre-reinforced composites. Finite Elem Anal Des 177:103418
Benedetto MF, Caggiano A, Etse G (2018) Virtual elements and zero thickness interface-based approach for fracture analysis of heterogeneous materials. Comput Methods Appl Mech Eng 338:41–67
Brezzi F, Lipnikov K, Simoncini V (2005) A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math Models Methods Appl Sci 15(10):1533–1551
Brezzi F, Marini LD (2013) Virtual element methods for plate bending problems. Comput Methods Appl Mech Eng 253:455–462
Chi H, Da Veiga LB, Paulino G (2017) Some basic formulations of the virtual element method (VEM) for finite deformations. Comput Methods Appl Mech Eng 318:148–192
Chi H, Pereira A, Menezes IF, Paulino GH (2020) Virtual element method (VEM)-based topology optimization: an integrated framework. Struct Multidiscip Optim 62(3):1089–1114
Chinosi C (2018) Virtual elements for the Reissner–Mindlin plate problem. Numer Methods Partial Differ Equ 34(4):1117–1144
Cihan M, Hudobivnik B, Aldakheel F, Wriggers P (2021) 3d mixed virtual element formulation for dynamic elasto-plastic analysis. Comput Mech 68:1
Da Veiga L, Beirao L, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A (2013) Basic principles of virtual element methods. Math Models Methods Appl Sci 23(01):199–214
Da Veiga L, Beirao L, Brezzi F, Marini LD, Russo A (2014) The hitchhiker’s guide to the virtual element method. Math Models Methods Appl Sci 24(08):1541–1573
Da Veiga LB, Brezzi F, Marini LD (2013) Virtual elements for linear elasticity problems. SIAM J Numer Anal 51(2):794–812
Da Veiga LB, Lipnikov K, Manzini G (2011) Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J Numer Anal 49(5):1737–1760
Da Veiga LB, Mora D, Rivera G (2019) Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates. Math Comput 88(315):149–178
Da Veiga LB, Russo A, Vacca G (2019) The virtual element method with curved edges. ESAIM Math Model Numer Anal 53(2):375–404
Dassault Systèmes: CATIA. https://www.3ds.com/products-services/catia/
Du X, Zhao G, Wang W (2015) Nitsche method for isogeometric analysis of Reissner–Mindlin plate with non-conforming multi-patches. Comput Aided Geometric Des 35:121–136
Du X, Zhao G, Wang W, Fang H (2020) Nitsche’s method for non-conforming multipatch coupling in hyperelastic isogeometric analysis. Comput Mech 65(3):687–710
Engwirda D (2014) Locally optimal Delaunay-refinement and optimisation-based mesh generation. Ph.D. Thesis, School of Mathematics and Statistics, The University of Sydney
Funken SA, Schmidt A (2020) Adaptive mesh refinement in 2D—an efficient implementation in matlab. Comput Methods Appl Math 20(3):459–479
Gain AL, Talischi C, Paulino GH (2014) On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput Methods Appl Mech Eng 282:132–160
Hudobivnik B, Aldakheel F, Wriggers P (2019) A low order 3D virtual element formulation for finite elasto-plastic deformations. Comput Mech 63(2):253–269
Hughes TJ (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Inc
Hussein A, Aldakheel F, Hudobivnik B, Wriggers P, Guidault PA, Allix O (2019) A computational framework for brittle crack-propagation based on efficient virtual element method. Finite Elem Anal Des 159:15–32
Kamel KEM, Sonon B, Massart TJ (2019) An integrated approach for the conformal discretization of complex inclusion-based microstructures. Comput Mech 64(4):1049–1071
Kim HJ, Seo YD, Youn SK (2010) Isogeometric analysis with trimming technique for problems of arbitrary complex topology. Comput Methods Appl Mech Eng 199(45–48):2796–2812
Knupp PM (2001) Algebraic mesh quality metrics. SIAM J Sci Comput 23(1):193–218
Liang B, Nagarajan A, Soghrati S (2019) Scalable parallel implementation of CISAMR: a non-iterative mesh generation algorithm. Comput Mech 64(1):173–195
Löhner R, Cebral JR, Camelli FE, Appanaboyina S, Baum JD, Mestreau EL, Soto OA (2008) Adaptive embedded and immersed unstructured grid techniques. Comput Methods Appl Mech Eng 197(25–28):2173–2197
Meng J, Mei L (2020) A linear virtual element method for the Kirchhoff plate buckling problem. Appl Math Lett 103:106188
Meng J, Mei L (2020) A mixed virtual element method for the vibration problem of clamped kirchhoff plate. Adv Comput Math 46(5):1–18
Mengolini M, Benedetto MF, Aragón AM (2019) An engineering perspective to the virtual element method and its interplay with the standard finite element method. Comput Methods Appl Mech Eng 350:995–1023
Mittal R, Iaccarino G (2005) Immersed boundary methods. Annu Rev Fluid Mech 37:239–261
Mora D, Velásquez I (2020) Virtual element for the buckling problem of Kirchhoff-Love plates. Comput Methods Appl Mech Eng 360:112687
Nagarajan A, Soghrati S (2018) Conforming to interface structured adaptive mesh refinement: 3D algorithm and implementation. Comput Mech 62(5):1213–1238
Nguyen-Thanh VM, Zhuang X, Nguyen-Xuan H, Rabczuk T, Wriggers P (2018) A virtual element method for 2D linear elastic fracture analysis. Comput Methods Appl Mech Eng 340:366–395
Ortiz-Bernardin A, Alvarez C, Hitschfeld-Kahler N, Russo A, Silva-Valenzuela R, Olate-Sanzana E (2019) Veamy: an extensible object-oriented C++ library for the virtual element method. Numer Algorithms 82(4):1189–1220
Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41(1):121–133
Piegl L, Tiller W (1996) The NURBS book, 2nd edn. Springer, Berlin
Rangarajan R, Lew AJ (2014) Universal meshes: a method for triangulating planar curved domains immersed in nonconforming meshes. Int J Numer Methods Eng 98(4):236–264
Roberts SA, Mendoza H, Brunini VE, Noble DR (2018) A verified conformal decomposition finite element method for implicit, many-material geometries. J Comput Phys 375:352–367
Rogers DF (2001) An introduction to NURBS: with historical perspective. Morgan Kaufmann
Ruess M, Schillinger D, Oezcan AI, Rank E (2014) Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Comput Methods Appl Mech Eng 269:46–71
Schillinger Dominik, Ruess Martin (2015) 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. https://doi.org/10.1007/s11831-014-9115-y
Schneiders R (1996) A grid-based algorithm for the generation of hexahedral element meshes. Eng Comput 12(3–4):168–177
Soghrati S, Nagarajan A, Liang B (2017) Conforming to interface structured adaptive mesh refinement: new technique for the automated modeling of materials with complex microstructures. Finite Elem Anal Des 125:24–40
Soghrati S, Xiao F, Nagarajan A (2017) A conforming to interface structured adaptive mesh refinement technique for modeling fracture problems. Comput Mech 59(4):667–684
Sutton OJ (2017) The virtual element method in 50 lines of MATLAB. Numer Algorithms 75(4):1141–1159
Talischi C, Paulino GH, Pereira A, Menezes IF (2012) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45(3):309–328
Van Huyssteen D, Reddy BD (2020) A virtual element method for isotropic hyperelasticity. Comput Methods Appl Mech Eng 367:113134
Wintiba B, Vasiukov D, Panier S, Lomov SV, Kamel KEM, Massart TJ (2020) Automated reconstruction and conformal discretization of 3D woven composite CT scans with local fiber volume fraction control. Compos Struct 248:112438
Wriggers P, Hudobivnik B (2017) A low order virtual element formulation for finite elasto-plastic deformations. Comput Methods Appl Mech Eng 327:459–477
Wriggers P, Hudobivnik B, Aldakheel F (2021) NURBS-based geometries: a mapping approach for virtual serendipity elements. Comput Methods Appl Mech Eng 378:113732
Wriggers P, Reddy B, Rust W, Hudobivnik B (2017) Efficient virtual element formulations for compressible and incompressible finite deformations. Comput Mech 60(2):253-268
Wriggers P, Rust WT, Reddy BD (2016) A virtual element method for contact. Comput Mech 58(6):1039–1050
Zhang J, Aragón AM (2022) An improved stress recovery technique for the unfitted finite element analysis of discontinuous gradient fields. Int J Numer Methods Eng 123(3):639–663
Zhang XS, Chi H, Paulino GH (2020) Adaptive multi-material topology optimization with hyperelastic materials under large deformations: a virtual element approach. Comput Methods Appl Mech Eng 370:112976
Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method: its basis and fundamentals, 7th edn. Elsevier, Amsterdam
Acknowledgements
The work is supported by the Project funded by China Postdoctoral Science Foundation (Project No. 2021M690294) and National Natural Science Foundation of China (Project Nos. 62102012, 52175213 and 61972011).
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Du, X., Wang, W., Zhao, G. et al. Virtual element method with adaptive refinement for problems of two-dimensional complex topology models from an engineering perspective. Comput Mech 70, 581–606 (2022). https://doi.org/10.1007/s00466-022-02179-6
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DOI: https://doi.org/10.1007/s00466-022-02179-6