zbMATH — the first resource for mathematics

A hybrid fundamental-solution-based 8-node element for axisymmetric elasticity problems. (English) Zbl 1464.74424
Summary: A novel hybrid finite element approach using fundamental solutions in conjunction with the boundary integration is proposed for the analysis of axisymmetric elasticity problems. In the formulation two displacement fields are independently assumed within the element domain and at the element boundary, which combines the advantages of conventional finite and boundary element methods. The resulting system of algebraic equations is symmetric and positive definite, which reduces the spatial dimension of the problem by one. It offers great flexibility in generating the mesh with arbitrary higher-order elements. Firstly, the proposed approach is validated by the analytical solution of a thick-walled hollow cylinder. A good agreement is observed. Meantime, the optimal number and location of source points for fundamental solutions as well as sensitivity to mesh distortion are studied. Finally, two more numerical examples including a flange and a wheel are provided. Compared to ABAQUS, the proposed approach exhibits higher efficiency, better convergence rate and strong capability to capture geometric singularities.

74S99 Numerical and other methods in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
74B05 Classical linear elasticity
Full Text: DOI
[1] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for axisymmetric potential problems, Int J Numer Methods Eng, 44, 1653-1669, (1999) · Zbl 0932.74078
[2] Provatidis, C., A boundary element method for axisymmetric potential problems with non-axisymmetric boundary conditions using fast fourier transform, Eng Comput, 15, 428-449, (1998) · Zbl 0924.65103
[3] Wang, K. Y.; Zhang, L. Q.; Li, P. C., A four-node hybrid-Trefftz annular element for analysis of axisymmetric potential problems, Finite Elem Anal Des, 60, 49-56, (2012)
[4] Provatidis, C.; Kanarachos, A., Performance of a macro-FEM approach using global interpolation (Coons’) functions in axisymmetric potential problems, Comput Struct, 79, 1769-1779, (2001)
[5] Zhou, J. C.; Wang, K. Y.; Li, P. C.; Miao, X. D., Hybrid fundamental solution based finite element method for axisymmetric potential problems, Eng Anal Bound Elem, 91, 82-91, (2018) · Zbl 1403.65149
[6] Zhou, J. C.; Wang, K. Y.; Li, P. C., Hybrid fundamental solution based finite element method for axisymmetric potential problems with arbitrary boundary conditions, Comput Struct, 212, 72-85, (2019)
[7] Provatidis, C., A fast Fourier-boundary element method for axisymmetric potential and elasticity problems with arbitrary boundary conditions, Comput Mech, 23, 258-270, (1999) · Zbl 0960.74072
[8] Jia, N.; Yao, Y.; Peng, Z. L.; Yang, Y. Z.; Chen, S. H., Surface effect in axisymmetric Hertzian contact problems, Int J Solids Struct, 150, 241-254, (2018)
[9] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for axisymmetric acoustic scattering and radiation problems, J Acoust Soc Am, 104, 3212-3218, (1998)
[10] Tsai, C. C.; Chen, C. S.; Hsu, T. W., The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations, Eng Anal Bound Elem, 33, 1396-1402, (2009) · Zbl 1244.65225
[11] Liu, G. H.; Zhang, Q. H.; Sze, K. Y., Spherical-wave based triangular finite element models for axial symmetric Helmholtz problems, Finite Elem Anal Des, 47, 342-350, (2011)
[12] Wang, M.; Watson, D.; Li, M., The method of particular solutions with polynomial basis functions for solving axisymmetric problems, Eng Anal Bound Elem, 90, 39-46, (2018) · Zbl 1403.65262
[13] Chen, J. F.; Zeng, S. B.; Dong, Q. Z.; Huang, Y. Q., Rapid simulation of electromagnetic telemetry using an axisymmetric semianalytical finite element method, J Appl Geophys, 137, 49-54, (2017)
[14] Sze, K. Y.; Wu, D., Transition finite element families for adaptive analysis of axisymmetric elasticity problems, Finite Elem Anal Des, 47, 360-372, (2011)
[15] Clough, R.; Rashid, Y., Finite element analysis of axisymmetric solids, ASCE Eng Mech Div J, 24, 117-123, (1965)
[16] Wan, D. T.; Hu, D. A.; Yang, G.; Long, T., A fully smoothed finite element method for analysis of axisymmetric problems, Eng Anal Bound Elem, 72, 78-88, (2016) · Zbl 1403.65142
[17] Godoy, G.; Boccardo, V.; Durán, M., A Dirichlet-to-Neumann finite element method for axisymmetric elastostatics in a semi-infinite domain, J Comput Phys, 328, 1-26, (2017)
[18] Fayed, M. H., Stresses at the intersection of sphere and cylinder by a variant finite-difference method, J Appl Mech, 41, 744-752, (1974) · Zbl 0292.73046
[19] Redekop, D.; Thompson, J. C., Use of fundamental solutions in the collocation method in axisymmetric elastostatics, Comput Struct, 17, 485-490, (1983)
[20] Mayr, M., On the numerical solution of axisymmetric elasticity problems using an integral equation approach, Mech Res Commun, 3, 393-398, (1976) · Zbl 0358.73026
[21] Kermanidis, T., A numerical solution for axially symmetrical elasticity problems, Int J Solids Struct, 11, 493-500, (1975) · Zbl 0296.73016
[22] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for axisymmetric elasticity problems, Comput Mech, 25, 524-532, (2000) · Zbl 1011.74005
[23] Cruse, T. A.; Snow, D. W.; Wilson, R. B., Numerical solutions in axisymmetric elasticity, Comput Struct, 7, 445-451, (1977) · Zbl 0359.73014
[24] Sladek, J.; Sladek, V.; Krivacek, J.; Zhang, C., Meshless local Petrov-Galerkin method for stress and crack analysis in 3-D axisymmetric FGM bodies, Comp Model Eng, 8, 259-270, (2005)
[25] Rice, J. S.; Mukherjee, S., Design sensitivity coefficients for axisymmetric elasticity problems by boundary element methods, Eng Anal Bound Elem, 7, 13-20, (1990)
[26] Rauchs, G., Direct-differentiation-based sensitivity analysis of an axisymmetric finite element formulation including torsion, Finite Elem Anal Des, 109, 65-72, (2016)
[27] Tokovyy, Y.; Ma, C. C., Analytical solutions to the axisymmetric elasticity and thermoelasticity problems for an arbitrarily inhomogeneous layer, Int J Eng Sci, 92, 1-17, (2015) · Zbl 1423.74111
[28] Jirousek, J.; Leon, N., A powerful finite element for plate bending, Comput Method Appl M, 12, 77-96, (1977) · Zbl 0366.73065
[29] Wang, K. Y.; Li, P. C.; Wang, D. Z., Trefftz-type FEM for solving orthotropic potential problems, Lat Am J Solid Struct, 11, 2537-2554, (2014)
[30] Jirousek, J.; Qin, Q. H., Application of hybrid-Trefftz element approach to transient heat conduction analysis, Comput Struct, 58, 195-201, (1996) · Zbl 0900.73802
[31] Dhanasekar, M.; Han, J. J.; Qin, Q. H., A hybrid-Trefftz element containing an elliptic hole, Finite Elem Anal Des, 42, 1314-1323, (2006)
[32] Qin, Q. H., Hybrid Trefftz finite-element approach for plate bending on an elastic foundation, Appl Math Model, 18, 334-339, (1994) · Zbl 0804.73070
[33] Qin, Q. H.; Wang, K. Y., Application of hybrid-Trefftz finite element method to frictional contact problems, Comput Assisted Mech Eng Sci, 15, 319-336, (2008) · Zbl 1421.74102
[34] Wang, K. Y.; Qin, Q. H.; Kang, Y. L.; Wang, J. S.; Qu, C. Y., A direct constraint-Trefftz FEM for analysing elastic contact problems, Int J Numer Meth Eng, 63, 1694-1718, (2005) · Zbl 1131.74341
[35] Qin, Q. H., Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach, Comput Mech, 31, 461-468, (2003) · Zbl 1038.74646
[36] Qin, Q. H., Variational formulations for TFEM of piezoelectricity, Int J Solids Struct, 40, 6335-6346, (2003) · Zbl 1057.74043
[37] Wang, H.; Qin, Q. H.; Xiao, Y., Special n-sided Voronoi fiber/matrix elements for clustering thermal effect in natural-hemp-fiber-filled cement composites, Int J Heat Mass Transf, 92, 228-235, (2016)
[38] Wang, H.; Zhao, X. J.; Wang, J. S., Interaction analysis of multiple coated fibers in cement composites by special n-sided interphase/fiber elements, Compos Sci Technol, 118, 117-126, (2015)
[39] Kompiš, V.; Búry, J., Hybrid-Trefffz finite element formulations based on the fundamental solution, (Mang, HA; Rammerstorfer, FG, Proceedings of the IUTAM symposium on discretization methods in structural mechanics, solid mechanics and its applications, (1999), Springer: Springer Dordrecht), 181-187
[40] Qin, Q. H.; Wang, H., MATLAB and C programming for Trefftz finite element methods, (2008), CRC: CRC Boca Raton
[41] Wang, H.; Qin, Q. H., A fundamental solution based FE model for thermal analysis of nanocomposites, WIT Trans Model Simul, 52, 191-202, (2011) · Zbl 1275.82010
[42] Cao, C. Y.; Qin, Q. H., Hybrid fundamental solution based finite element method: theory and applications, Adv Math Phys, 2015, 1-38, (2015)
[43] Qin, Q. H., Trefftz finite and boundary element method, Appl Mech Rev, 54, B99-B100, (2001)
[44] Wang, H.; Qin, Q. H., Fundamental-solution-based finite element model for plane orthotropic elastic bodies, Eur J Mech A Solid, 29, 801-809, (2010)
[45] Hirshikesh, N. S.; Annabattula, R. K.; Bordas, S.; Atroshchenko, E., Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties, Asia Pac J Comput Engin, 4, 3-17, (2017)
[46] Reismann, H.; Pawlik, P. S., Elasticity, theory and applications, (1980), Wiley: Wiley New York · Zbl 0515.73009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.