[en] A novel approach of polygonal scaled boundary isogeometric analysis is proposed for 2D elasticity problems involving trimmed geometries. The method addresses the challenge of efficiently handling trimmed geometries directly within the analysis process. It employs the Newton-Raphson method to search for intersection points between the trimming curve and isoparametric curves of the NURBS surface. The approach involves mapping untrimmed internal grids bounded by isoparametric curve segments and trimmed elements bounded by trimming curve and isoparametric curve segments into scaled boundary elements. Field variable approximations are achieved using NURBS basis functions. The system equation is derived through the virtual work statement, and a hybrid variable is introduced in the solution procedure. The method preserves the dimension reduction and semi-analytical characteristics of the classical SBFEM. Refinement strategies (h−, k−, p−refinements) for IGA are applied to implement high-order continuity boundary elements at the polygonal element level. The approach is capable of handling arbitrary complex problem domains without the necessity of sub-domain division. It accurately represents curved elements with few control points, thereby enhancing computational accuracy compared to SBFEM. The proposed method has been demonstrated to be correct, accurate, and efficient. It holds the potential to advance IGA-based numerical methods and provide valuable guidance for the development of large-scale integration software in the framework of IGA and SBFEM.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others
Author, co-author :
Zang, Quansheng; School of Water Conservancy and Transportation, Zhengzhou University, Zhengzhou, China ; National Local Joint Engineering Laboratory of Major Infrastructure Testing and Rehabilitation Technology, Zhengzhou, China ; Department of Hydraulic Engineering, School of Infrastructure Engineering, Dalian University of Technology, Dalian, China ; Department of Engineering, Faculty of Science, Technology and Medicine, University of Luxembourg, Luxembourg
JANSARI, Chintankumar Vipulbhai ; University of Luxembourg > Faculty of Science, Technology and Medicine > Department of Engineering > Team Stéphane BORDAS
Bordas, Stéphane P.A. ; Department of Engineering, Faculty of Science, Technology and Medicine, University of Luxembourg, Luxembourg
Liu, Jun; Department of Hydraulic Engineering, School of Infrastructure Engineering, Dalian University of Technology, Dalian, China
External co-authors :
yes
Language :
English
Title :
Trimming with polygonal scaled boundary isogeometric method
The authors gratefully acknowledge the Legato team work, the first author also acknowledges the support from the School of Water Conservancy and Transportation, Zhengzhou University . This research was supported by Grant 52179124 from the National Natural Science Foundation of China for which the first and the fourth authors are grateful.
Hughes, T.J., Reali, A., Sangalli, G., Efficient quadrature for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 199 (2010), 301–313.
Cottrell, J., Hughes, T., Reali, A., Studies of refinement and continuity in isogeometric structural analysis. Comput Methods Appl Mech Eng 196 (2007), 4160–4183.
Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J., Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195 (2006), 5257–5296.
Hughes, T.J., Reali, A., Sangalli, G., Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS. Comput Methods Appl Mech Eng 197 (2008), 4104–4124.
Bazilevs, Y., Calo, V., Cottrell, J., Hughes, T., Reali, A., Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197 (2007), 173–201.
Nguyen, V.P., Anitescu, C., Bordas, S.P., Rabczuk, T., Isogeometric analysis: an overview and computer implementation aspects. Math Comput Simul 117 (2015), 89–116.
Bordas, S., Menk, A., Natarajan, S., Partition of unity methods. first ed., 2023, Wiley.
Hu, Q., Chouly, F., Hu, P., Cheng, G., Bordas, S.P., Skew-symmetric Nitsche's formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact. Comput Methods Appl Mech Eng 341 (2018), 188–220.
Atroshchenko, E., Tomar, S., Xu, G., Bordas, S.P., Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub-and super-geometric analysis to Geometry-Independent Field approximaTion (GIFT). Int J Numer Methods Eng 114 (2018), 1131–1159.
Marussig, B., Zechner, J., Beer, G., Fries, T.-P., Fast isogeometric boundary element method based on independent field approximation. Comput Methods Appl Mech Eng 284 (2015), 458–488.
Phung-Van, P., Thai, C.H., A novel size-dependent nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite nanoplates. Eng Comput, 2021, 1–14.
Yu, P., Anitescu, C., Tomar, S., Bordas, S.P.A., Kerfriden, P., Adaptive isogeometric analysis for plate vibrations: an efficient approach of local refinement based on hierarchical a posteriori error estimation. Comput Methods Appl Mech Eng 342 (2018), 251–286.
Videla, J., Anitescu, C., Khajah, T., Bordas, S.P., Atroshchenko, E., h- and p-adaptivity driven by recovery and residual-based error estimators for PHT-splines applied to time-harmonic acoustics. Comput Math Appl 77 (2019), 2369–2395.
Ding, C., Deokar, R.R., Lian, H., Ding, Y., Li, G., Cui, X., et al. Resolving high frequency issues via proper orthogonal decomposition based dynamic isogeometric analysis for structures with dissimilar materials. Comput Methods Appl Mech Eng, 359, 2020, 112753.
Ding, C., Tamma, K.K., Cui, X., Ding, Y., Li, G., Bordas, S.P., An nth high order perturbation-based stochastic isogeometric method and implementation for quantifying geometric uncertainty in shell structures. Adv Eng Softw, 148, 2020, 102866.
Jansari, C., Videla, J., Natarajan, S., Bordas, S.P., Atroshchenko, E., Adaptive enriched geometry independent field approximation for 2D time-harmonic acoustics. Comput Struct, 263, 2022, 106728.
Yu, P., Bordas, S.P.A., Kerfriden, P., Adaptive isogeometric analysis for transient dynamics: space–time refinement based on hierarchical a-posteriori error estimations. Comput Methods Appl Mech Eng, 394, 2022, 114774.
Jiang, K., Zhu, X., Hu, C., Hou, W., Hu, P., Bordas, S.P., An enhanced extended isogeometric analysis with strong imposition of essential boundary conditions for crack problems using B++ splines. Appl Math Model 116 (2023), 393–414.
Simpson, R.N., Bordas, S.P., Trevelyan, J., Rabczuk, T., A two-dimensional isogeometric boundary element method for elastostatic analysis. Comput Methods Appl Mech Eng 209 (2012), 87–100.
Zang, Q., Liu, J., Ye, W., Lin, G., Isogeometric boundary element for analyzing steady-state heat conduction problems under spatially varying conductivity and internal heat source. Comput Math Appl 80 (2020), 1767–1792.
Chen, L., Lu, C., Lian, H., Liu, Z., Zhao, W., Li, S., et al. Acoustic topology optimization of sound absorbing materials directly from subdivision surfaces with isogeometric boundary element methods. Comput Methods Appl Mech Eng, 362, 2020, 112806.
Chen, L., Zhang, Y., Lian, H., Atroshchenko, E., Ding, C., Bordas, S.P., Seamless integration of computer-aided geometric modeling and acoustic simulation: Isogeometric boundary element methods based on Catmull-Clark subdivision surfaces. Adv Eng Softw, 149, 2020, 102879.
Chen, L., Lian, H., Natarajan, S., Zhao, W., Chen, X., Bordas, S., Multi-frequency acoustic topology optimization of sound-absorption materials with isogeometric boundary element methods accelerated by frequency-decoupling and model order reduction techniques. Comput Methods Appl Mech Eng, 395, 2022, 114997.
Ghorashi, S.S., Valizadeh, N., Mohammadi, S., Extended isogeometric analysis for simulation of stationary and propagating cracks. Int J Numer Methods Eng 89 (2012), 1069–1101.
Peake, M.J., Trevelyan, J., Coates, G., Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems. Comput Methods Appl Mech Eng 259 (2013), 93–102.
Xue, B., Lin, G., Hu, Z., Scaled boundary isogeometric analysis for electrostatic problems. Eng Anal Bound Elem 85 (2017), 20–29.
Zang, Q., Liu, J., Ye, W., Yang, F., Hao, C., Lin, G., Static and free vibration analyses of functionally graded plates based on an isogeometric scaled boundary finite element method. Compos Struct, 288, 2022, 115398.
Zang, Q., Liu, J., Ye, W., Yang, F., Pang, R., Lin, G., High-performance bending and buckling analyses of cylindrical shells resting on elastic foundation using isogeometric scaled boundary finite element method. Eur J Mech A, Solids, 2023, 105013.
Litke, N., Levin, A., Schröder, P., Trimming for subdivision surfaces. Comput Aided Geom Des 18 (2001), 463–481.
Seo, Y.-D., Kim, H.-J., Youn, S.-K., Isogeometric topology optimization using trimmed spline surfaces. Comput Methods Appl Mech Eng 199 (2010), 3270–3296.
Kim, H.-J., Seo, Y.-D., Youn, S.-K., Isogeometric analysis for trimmed CAD surfaces. Comput Methods Appl Mech Eng 198 (2009), 2982–2995.
Kim, H.-J., Seo, Y.-D., Youn, S.-K., Isogeometric analysis with trimming technique for problems of arbitrary complex topology. Comput Methods Appl Mech Eng 199 (2010), 2796–2812.
Schmidt, R., Wüchner, R., Bletzinger, K.-U., Isogeometric analysis of trimmed NURBS geometries. Comput Methods Appl Mech Eng 241 (2012), 93–111.
Beer, G., Marussig, B., Zechner, J., Dünser, C., Fries, T.-P., Boundary element analysis with trimmed NURBS and a generalized IGA approach. arXiv preprint arXiv:1406.3499, 2014.
Beer, G., Marussig, B., Zechner, J., A simple approach to the numerical simulation with trimmed CAD surfaces. Comput Methods Appl Mech Eng 285 (2015), 776–790.
Ruess, M., Schillinger, D., Oezcan, A.I., Rank, E., Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Comput Methods Appl Mech Eng 269 (2014), 46–71.
Wang, Y., Benson, D.J., Nagy, A.P., A multi-patch nonsingular isogeometric boundary element method using trimmed elements. Comput Mech 56 (2015), 173–191.
Nagy, A.P., Benson, D.J., On the numerical integration of trimmed isogeometric elements. Comput Methods Appl Mech Eng 284 (2015), 165–185.
Marussig, B., Hughes, T.J., A review of trimming in isogeometric analysis: challenges, data exchange and simulation aspects. Arch Comput Methods Eng 25 (2018), 1059–1127.
Leidinger, L., Breitenberger, M., Bauer, A., Hartmann, S., Wüchner, R., Bletzinger, K.-U., et al. Explicit dynamic isogeometric B-Rep analysis of penalty-coupled trimmed NURBS shells. Comput Methods Appl Mech Eng 351 (2019), 891–927.
Antolin, P., Buffa, A., Puppi, R., Wei, X., Overlapping multipatch isogeometric method with minimal stabilization. SIAM J Sci Comput 43 (2021), A330–A354.
Coradello, L., D'Angella, D., Carraturo, M., Kiendl, J., Kollmannsberger, S., Rank, E., et al. Hierarchically refined isogeometric analysis of trimmed shells. Comput Mech 66 (2020), 431–447.
Pasch, T., Leidinger, L., Apostolatos, A., Wüchner, R., Bletzinger, K.-U., Duddeck, F., A priori penalty factor determination for (trimmed) NURBS-based shells with Dirichlet and coupling constraints in isogeometric analysis. Comput Methods Appl Mech Eng, 377, 2021, 113688.
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 (2002), 183–197.
Wolf, J.P., Song, C., The scaled boundary finite-element method–a fundamental solution-less boundary-element method. Comput Methods Appl Mech Eng 190 (2001), 5551–5568.
Deeks, A.J., Wolf, J.P., A virtual work derivation of the scaled boundary finite-element method for elastostatics. Comput Mech 28 (2002), 489–504.
Doherty, J., Deeks, A., Adaptive coupling of the finite-element and scaled boundary finite-element methods for non-linear analysis of unbounded media. Comput Geotech 32 (2005), 436–444.
Vu, T.H., Deeks, A.J., Use of higher-order shape functions in the scaled boundary finite element method. Int J Numer Methods Eng 65 (2006), 1714–1733.
Man, H., Song, C., Gao, W., Tin-Loi, F., A unified 3D-based technique for plate bending analysis using scaled boundary finite element method. Int J Numer Methods Eng 91 (2012), 491–515.
Man, H., Song, C., Xiang, T., Gao, W., Tin-Loi, F., High-order plate bending analysis based on the scaled boundary finite element method. Int J Numer Methods Eng 95 (2013), 331–360.
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 (2019), 563–581.
Dölling, S., Hahn, J., Felger, J., Bremm, S., Becker, W., A scaled boundary finite element method model for interlaminar failure in composite laminates. Compos Struct, 241, 2020, 111865.
Ye, N., Su, C., Yang, Y., PSBFEM-Abaqus: development of user element subroutine (UEL) for polygonal scaled boundary finite element method in Abaqus. Math Probl Eng, 2021, 2021.
Zhang, Y., Lin, G., Hu, Z., Isogeometric analysis based on scaled boundary finite element method. IOP conference series: materials science and engineering, vol. 10, 2010, IOP Publishing, 012237.
Natarajan, S., Wang, J., Song, C., Birk, C., Isogeometric analysis enhanced by the scaled boundary finite element method. Comput Methods Appl Mech Eng 283 (2015), 733–762.
Li, P., Liu, J., Lin, G., Zhang, P., Yang, G., A NURBS-based scaled boundary finite element method for the analysis of heat conduction problems with heat fluxes and temperatures on side-faces. Int J Heat Mass Transf 113 (2017), 764–779.
Chasapi, M., Klinkel, S., A scaled boundary isogeometric formulation for the elasto-plastic analysis of solids in boundary representation. Comput Methods Appl Mech Eng 333 (2018), 475–496.
Arioli, C., Shamanskiy, A., Klinkel, S., Simeon, B., Scaled boundary parametrizations in isogeometric analysis. Comput Methods Appl Mech Eng 349 (2019), 576–594.
Zang, Q., Liu, J., Ye, W., Gao, H., Lin, G., Plate-bending analysis by NURBS-based scaled boundary finite-element method. J Eng Mech, 147, 2021, 04021054.
Natarajan, S., Bordas, S., Roy Mahapatra, D., Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping. Int J Numer Methods Eng 80 (2009), 103–134.
Nguyen-Xuan, H., A polygonal finite element method for plate analysis. Comput Struct 188 (2017), 45–62.
Francis, A., Ortiz-Bernardin, A., Bordas, S.P., Natarajan, S., Linear smoothed polygonal and polyhedral finite elements. Int J Numer Methods Eng 109 (2017), 1263–1288.
She, Z., Wang, K., Li, P., Hybrid Trefftz polygonal elements for heat conduction problems with inclusions/voids. Comput Math Appl 78 (2019), 1978–1992.
Chen, K., Zou, D., Kong, X., Chan, A., Hu, Z., A novel nonlinear solution for the polygon scaled boundary finite element method and its application to geotechnical structures. Comput Geotech 82 (2017), 201–210.
Yang, Y., Zhang, Z., Feng, Y., Yu, Y., Wang, K., Liang, L., A polygonal scaled boundary finite element method for solving heat conduction problems. arXiv preprint arXiv:2106.12283, 2021.
Yu, B., Hu, P., Saputra, A.A., Gu, Y., The scaled boundary finite element method based on the hybrid quadtree mesh for solving transient heat conduction problems. Appl Math Model 89 (2021), 541–571.
Hormann, K., Agathos, A., The point in polygon problem for arbitrary polygons. Comput Geom 20 (2001), 131–144.
Zang, Q., Bordas, S.P., Liu, J., Natarajan, S., NURBS-enhanced polygonal scaled boundary finite element method for heat diffusion in anisotropic media with internal heat sources. Eng Anal Bound Elem 148 (2023), 279–292.
Gould, P.L., Introduction to linear elasticity. third ed., 1994, Springer, New York, USA.
Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F., Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45 (2012), 309–328.
Song, C., Scaled boundary finite element method: introduction to theory and implementation. 2018, John Wiley and Sons, Hoboken, NJ, USA.
