Stable generalized iso-geometric analysis (SGIGA) for problems with discontinuities and singularities. (English) Zbl 1440.65198
Summary: Numerical analysis of physical/mathematical problems based on generalized/extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices. Blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks. The presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed. In order to alleviate these drawbacks, three different Stable Generalized IsoGeometric Analysis (SGIGA) methods are proposed in this paper. In SGIGA, the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function. Problems with weak and strong discontinuities, singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed SGIGA methods. From the results, it is observed that SGIGA yields optimal convergence rates as well as better conditioning of system matrices. The results obtained from the proposed SGIGA methods are also compared with the results from the established methods, Stable Generalized Finite Element Method (SGFEM) and eXtended IsoGeometric Analysis (XIGA), to study the relative performances with respect to accuracy and conditioning.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65D07 | Numerical computation using splines |
Keywords:
finite element method; stable generalized isogeometric analysis; discontinuities; singularities; blending elements; ill conditioningSoftware:
XFEMReferences:
| [1] | Melenk, J. M.; Babuška, I., The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139, 1, 289-314 (1996) · Zbl 0881.65099 |
| [2] | Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the generalized finite element method, Comput. Methods Appl. Mech. Engrg., 181, 1, 43-69 (2000) · Zbl 0983.65127 |
| [3] | Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, Internat. J. Numer. Methods Engrg., 45, 5, 601-620 (1999) · Zbl 0943.74061 |
| [4] | Belytschko, T.; Moës, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg., 50, 4, 993-1013 (2001) · Zbl 0981.74062 |
| [5] | Sukumar, N.; Huang, Z. Y.; Prévost, J.-H.; Suo, Z., Partition of unity enrichment for bimaterial interface cracks, Internat. J. Numer. Methods Engrg., 59, 8, 1075-1102 (2004) · Zbl 1041.74548 |
| [6] | Bayesteh, H.; Mohammadi, S., XFEM Fracture analysis of orthotropic functionally graded materials, Composites B, 44, 1, 8-25 (2013) |
| [7] | Chessa, J.; Belytschko, T., An extended finite element method for two-phase fluids, J. Appl. Mech., 70, 1, 10-17 (2003) · Zbl 1110.74391 |
| [8] | Gerstenberger, A.; Wall, W. A., An extended finite element method/Lagrange multiplier based approach for fluid – structure interaction, Comput. Methods Appl. Mech. Engrg., 197, 19, 1699-1714 (2008) · Zbl 1194.76117 |
| [9] | Bordas, S.; Nguyen, P. V.; Dunant, C.; Guidoum, A.; Dang, H. N., An extended finite element library, Internat. J. Numer. Methods Engrg., 71, 703-732 (2007) · Zbl 1194.74367 |
| [10] | Chessa, J.; Wang, H.; Belytschko, T., On the construction of blending elements for local partition of unity enriched finite elements, Internat. J. Numer. Methods Engrg., 57, 7, 1015-1038 (2003) · Zbl 1035.65122 |
| [11] | Gracie, R.; Wang, H.; Belytschko, T., Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods, Internat. J. Numer. Methods Engrg., 74, 11, 1645-1669 (2008) · Zbl 1195.74175 |
| [12] | Tarancón, J. E.; Vercher, A.; Giner, E.; Fuenmayor, F. J., Enhanced blending elements for XFEM applied to linear elastic fracture mechanics, Internat. J. Numer. Methods Engrg., 77, 1, 126-148 (2009) · Zbl 1195.74199 |
| [13] | Fries, T.-P., A corrected XFEM approximation without problems in blending elements, Int. J. Numer. Methods Eng., 75, 5, 503-532 (2008) · Zbl 1195.74173 |
| [14] | Béchet, E.; Minnebo, H.; Moës, N.; Burgardt, B., Improved implementation and robustness study of the X-FEM for stress analysis around cracks, Internat. J. Numer. Methods Engrg., 64, 8, 1033-1056 (2005) · Zbl 1122.74499 |
| [15] | Menk, A.; Bordas, S. P.A., A robust preconditioning technique for the extended finite element method, Internat. J. Numer. Methods Engrg., 85, 13, 1609-1632 (2011) · Zbl 1217.74128 |
| [16] | Agathos, K.; Bordas, S. P.A.; Chatzi, E., Improving the conditioning of XFEM/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization, Comput. Methods Appl. Mech. Engrg., 346, 1051-1073 (2019) · Zbl 1440.74351 |
| [17] | Babuška, I.; Banerjee, U., Stable generalized finite element method (SGFEM), Comput. Methods Appl. Mech. Engrg., 201-204, 91-111 (2012) · Zbl 1239.74093 |
| [18] | Sauerland, H.; Fries, T.-P., The stable XFEM for two-phase flows, Comput. & Fluids, 87, 41-49 (2013) · Zbl 1290.76073 |
| [19] | Gupta, V.; Duarte, C. A.; Babuška, I.; Banerjee, U., A stable and optimally convergent Generalized FEM (sgfem) for linear elastic fracture mechanics, Comput. Methods Appl. Mech. Engrg., 266, 23-39 (2013) · Zbl 1286.74102 |
| [20] | Gupta, V.; Duarte, C. A.; Babuška, I.; Banerjee, U., Stable GFEM (sgfem): Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics, Comput. Methods Appl. Mech. Engrg., 289, 355-386 (2015) · Zbl 1423.74886 |
| [21] | Zhang, Q.; Banerjee, U.; Babuška, I., Higher order stable generalized finite element method, Numer. Math., 128, 1, 1-29 (2014) · Zbl 1304.65254 |
| [22] | Zhang, Q.; Babuška, I.; Banerjee, U., Robustness in stable generalized finite element methods (SGFEM) applied to poisson problems with crack singularities, Comput. Methods Appl. Mech. Engrg., 311, 476-502 (2016) · Zbl 1439.74479 |
| [23] | Kergrene, K.; Babuška, I.; Banerjee, U., Stable generalized finite element method and associated iterative schemes; application to interface problems, Comput. Methods Appl. Mech. Engrg., 305, 1-36 (2016) · Zbl 1425.74472 |
| [24] | Agathos, K.; Chatzi, E.; Bordas, S. P.A., Stable 3D extended finite elements with higher order enrichment for accurate non planar fracture, Comput. Methods Appl. Mech. Engrg., 306, 19-46 (2016) · Zbl 1436.74064 |
| [25] | Agathos, K.; Ventura, G.; Chatzi, E.; Bordas, S. P.A., Stable 3D XFEM/vector level sets for non-planar 3D crack propagation and comparison of enrichment schemes, Int. J. Numer. Methods Eng., 113, 252-276 (2018) · Zbl 07867265 |
| [26] | Ventura, G.; Gracie, R.; Belytschko, T., Fast integration and weight function blending in the extended finite element method, Int. J. Numer. Methods Eng., 77, 1-29 (2009) · Zbl 1195.74201 |
| [27] | 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, 39, 4135-4195 (2005) · Zbl 1151.74419 |
| [28] | Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Engrg., 195, 41, 5257-5296 (2006) · Zbl 1119.74024 |
| [29] | Cottrell, J. A.; Hughes, T. J.R.; Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Comput. Methods Appl. Mech. Engrg., 196, 41, 4160-4183 (2007) · Zbl 1173.74407 |
| [30] | Benson, D. J.; Bazilevs, Y.; Hsu, M.-C.; Hughes, T. J.R., Isogeometric shell analysis: the Reissner-Mindlin shell, Comput. Methods Appl. Mech. Engrg., 199, 5, 276-289 (2010) · Zbl 1227.74107 |
| [31] | Benson, D. J.; Bazilevs, Y.; De Luycker, E.; Hsu, M.-C.; Scott, M.; Hughes, T. J.R.; Belytschko, T., A generalized finite element formulation for arbitrary basis functions: from isogeometric analysis to XFEM, Internat. J. Numer. Methods Engrg., 83, 6, 765-785 (2010) · Zbl 1197.74177 |
| [32] | De Luycker, E.; Benson, D. J.; Belytschko, T.; Bazilevs, Y.; Hsu, M.-C., X-FEM in isogeometric analysis for linear fracture mechanics, Internat. J. Numer. Methods Engrg., 87, 6, 541-565 (2011) · Zbl 1242.74105 |
| [33] | Ghorashi, S. S.; Valizadeh, N.; Mohammadi, S., Extended isogeometric analysis for simulation of stationary and propagating cracks, Internat. J. Numer. Methods Engrg., 89, 9, 1069-1101 (2012) · Zbl 1242.74119 |
| [34] | Bhardwaj, G.; Singh, I. V.; Mishra, B. K.; Bui, T. Q., Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions, Compos. Struct., 126, 347-359 (2015) |
| [36] | Nguyen, V. P.; Anitescu, C.; Bordas, S. P.A.; Rabczuk, T., Isogeometric analysis: An overview and computer implementation aspects, Math. Comput. Simulation, 117, 89-116 (2015) · Zbl 1540.65492 |
| [37] | Dinachandra, M.; Raju, Sethuraman, Plane wave enriched partition of unity isogeometric analysis (PUIGA) for 2D-Helmholtz problems, Comput. Methods Appl. Mech. Engrg., 335, 380-402 (2018) · Zbl 1440.74389 |
| [38] | Popov, E. P., Engineering Mechanics of Solids (1998), Prentice-Hall |
| [39] | Piegl, L.; Tiller, W., The NURBS Book (Monographs in Visual Communication) (1996), Springer-Verlag |
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