Journal of Applied and Computational Mechanics
Accuracy and Convergence Rate Comparative Investigation on Polytope Smoothed and Scaled Boundary Finite Element
Document Type : Research Paper
Authors
Department of Civil Engineering, Rajamangala University of Technology Thanyaburi, Pathumthani, 12110, Thailand
Abstract
Continuity and discontinuity of two-dimensional domains are thoroughly investigated for accuracy and convergence rate using two prominent discretization methods, namely smoothed and scaled boundary finite element. Because of their capability and versatility when compared to primitive elements, N-sided polygonal elements discretized from modified DistMesh and PolyMesher schemes are used. In terms of accuracy and convergence rate, NSFEM and SBFEM are found to be superior to CSFEM and ESFEM regardless of meshing alternative. The best accuracy occurs at NSFEM and SBFEM, and the obtained convergence rates are optimal. Particularly, in the smoothing domain, it is believed that DistMesh has more promising potential than PolyMesher does; yet, in the discontinuity domain, PolyMesher has been discovered to be more powerful while maintaining its efficiency.
Keywords
Main Subjects
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[1] Bathe, K.J., Finite element method, Wiley encyclopedia of computer science and engineering, 2007.
[2] Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z., The finite element method: its basis and fundamentals, Elsevier, 2005.
[3] Hughes, T.J., The finite element method: linear static and dynamic finite element analysis, Courier Corporation, 2012.
[4] Liu, G., Zhang, G., A normed G space and weakened weak (W2) formulation of a cell-based smoothed point interpolation method, International Journal of Computational Methods, 6(1), 2009, 147-179.
[5] Fawkes, A., Owen, D., Luxmoore, A., An assessment of crack tip singularity models for use with isoparametric elements, Engineering Fracture Mechanics, 11(1), 1979, 143-159.
[6] Medina, F., Taylor, R.L., Finite element techniques for problems of unbounded domains, International Journal for Numerical Methods in Engineering, 19(8), 1983, 1209-1226.
[7] Liu, G.-R., Trung, N., Smoothed finite element methods, CRC press, 2006.
[8] Banerjee, P.K., Butterfield, R., Boundary element methods in engineering science, McGraw-Hill (UK), 1981.
[9] Song, C., Wolf, J.P., The scaled boundary finite-element method—alias consistent infinitesimal finite-element cell method—for elastodynamics, Computer Methods in Applied Mechanics and Engineering, 147(3-4), 1997, 329-355.
[10] Wolf, J.P., The scaled boundary finite element method, John Wiley & Sons, 2003.
[11] Belytschko, T., Black, T., Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, 45(5), 1999, 601-620.
[12] Surendran, M., et al., Linear smoothed extended finite element method for fatigue crack growth simulations, Engineering Fracture Mechanics, 206, 2019, 551-564.
[13] De Almeida, J.M., De Freitas, J.T., Alternative approach to the formulation of hybrid equilibrium finite elements, Computers & Structures, 40(4), 1991, 1043-1047.
[14] Pian, T.H., Wu, C.-C., Hybrid and incompatible finite element methods, CRC press, 2005.
[15] Zeng, W., Liu, G., Smoothed finite element methods (S-FEM): an overview and recent developments, Archives of Computational Methods in Engineering, 25(2), 2018, 397-435.
[16] Liu, G.-R., The smoothed finite element method (S-FEM): A framework for the design of numerical models for desired solutions, Frontiers of Structural and Civil Engineering, 13(2), 2019, 456-477.
[17] He, Z., et al., An improved modal analysis for three-dimensional problems using face-based smoothed finite element method, Acta Mechanica Solida Sinica, 26(2), 2013, 140-150.
[18] Li, E., et al., Smoothed finite element method for analysis of multi-layered systems–Applications in biomaterials, Computers & Structures, 168, 2016, 16-29.
[19] Nguyen-Thoi, T., et al., A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes, Computer Methods in Applied Mechanics and Engineering, 199(45-48), 2010, 3005-3027.
[20] Ya, S., et al., An open-source ABAQUS implementation of the scaled boundary finite element method to study interfacial problems using polyhedral meshes, Computer Methods in Applied Mechanics and Engineering, 381, 2021, 113766.
[21] Greaves, D.M., Borthwick, A., Hierarchical tree‐based finite element mesh generation, International Journal for Numerical Methods in Engineering, 45(4), 1999, 447-471.
[22] Liu, Y., et al., Automatic polyhedral mesh generation and scaled boundary finite element analysis of STL models, Computer Methods in Applied Mechanics and Engineering, 313, 2017, 106-132.
[23] Huang, Y., et al., 3D meso-scale fracture modelling and validation of concrete based on in-situ X-ray Computed Tomography images using damage plasticity model, International Journal of Solids and Structures, 67, 2015, 340-352.
[24] Zhang, J., Song, C., A polytree based coupling method for non-matching meshes in 3D, Computer Methods in Applied Mechanics and Engineering, 349, 2019, 743-773.
[25] Fix, G.J., A Rational Finite Element Basis (Eugene L. Wachpress), Society for Industrial and Applied Mathematics, 1978.
[26] Nguyen-Xuan, H., et al., A polytree-based adaptive approach to limit analysis of cracked structures, Computer Methods in Applied Mechanics and Engineering, 313, 2017, 1006-1039.
[27] Persson, P.-O., Strang, G., A simple mesh generator in MATLAB, SIAM Review, 46(2), 2004, 329-345.
[28] Talischi, C., et al., PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Structural and Multidisciplinary Optimization, 45(3), 2012, 309-328.
[29] Talischi, C., et al., Polygonal finite elements for topology optimization: A unifying paradigm, International Journal for Numerical Methods in Engineering, 82(6), 2010, 671-698.
[30] Manual, A.S.U.S., Abaqus 6.11. http://130.149, 2012. 89(2080): p. v6.
[31] Liu, G., et al., A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, International Journal of Computational Methods, 2(4), 2005, 645-665.
[32] Liu, G., et al., A linearly conforming radial point interpolation method for solid mechanics problems, International Journal of Computational Methods, 3(4), 2006, 401-428.
[33] Wolf, J.P., Song, C., Finite-element modelling of unbounded media, Wiley Chichester, 1996.
[34] Deeks, A.J., Wolf, J.P., A virtual work derivation of the scaled boundary finite-element method for elastostatics, Computational Mechanics, 28(6), 2002, 489-504.
[35] Song, C., A matrix function solution for the scaled boundary finite-element equation in statics, Computer Methods in Applied Mechanics and Engineering, 193(23-26), 2004, 2325-2356.
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