Please wait a minute...

Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (1) : 201-214     https://doi.org/10.1007/s11709-018-0488-0
RESEARCH ARTICLE |
Scaled boundary finite element method with exact defining curves for two-dimensional linear multi-field media
Jaroon RUNGAMORNRAT1, Chung Nguyen VAN1,2()
1. Applied Mechanics and Structures Research Unit, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand
2. Faculty of Civil Engineering, Ho Chi Minh City of Technology and Education, Ho Chi Minh 721400, Vietnam
Download: PDF(489 KB)   HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract

This paper presents an efficient and accurate numerical technique based upon the scaled boundary finite element method for the analysis of two-dimensional, linear, second-order, boundary value problems with a domain completely described by a circular defining curve. The scaled boundary finite element formulation is established in a general framework allowing single-field and multi-field problems, bounded and unbounded bodies, distributed body source, and general boundary conditions to be treated in a unified fashion. The conventional polar coordinates together with a properly selected scaling center are utilized to achieve the exact description of the circular defining curve, exact geometry of the domain, and exact spatial differential operators. Standard finite element shape functions are employed in the discretization of both trial and test functions in the circumferential direction and the resulting eigenproblem is solved by a selected efficient algorithm. The computational performance of the implemented procedure is then fully investigated for various scenarios to demonstrate the accuracy in comparison with standard linear elements.

Keywords multi-field problems      defining curve      exact geometry      general boundary conditions      SBFEM     
Corresponding Authors: Chung Nguyen VAN   
Just Accepted Date: 14 May 2018   Online First Date: 27 June 2018    Issue Date: 04 January 2019
 Cite this article:   
Jaroon RUNGAMORNRAT,Chung Nguyen VAN. Scaled boundary finite element method with exact defining curves for two-dimensional linear multi-field media[J]. Front. Struct. Civ. Eng., 2019, 13(1): 201-214.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-018-0488-0
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I1/201
Service
E-mail this article
E-mail Alert
RSS
Articles by authors
Jaroon RUNGAMORNRAT
Chung Nguyen VAN
Fig.1  Schematic of two-dimensional, multi-field body subjected to external excitations
Fig.2  Schematic of generic body Ω, corresponding boundary, and its approximation Ωh
Fig.3  Schematic of scaling center and defining curve of (a) 2-node isoparametric linear element and (b) 2-node circular-arc element
Fig.4  Schematic of a quarter of a ring subjected to non-uniform heat source and mixed boundary conditions
n u1/u1exact
θ/π=0.1 θ/π=0.2 θ/π=0.3 θ/π=0.4
Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2
2 1.0609 1.0000 1.1218 1.0000 1.1483 1.0000 1.1405 1.0000
4 1.0252 1.0000 1.0336 1.0000 1.0352 1.0000 1.0354 1.0000
8 1.0067 1.0000 1.0085 1.0000 1.0088 1.0000 1.0089 1.0000
16 1.0017 1.0000 1.0021 1.0000 1.0022 1.0000 1.0022 1.0000
Tab.1  Normalized temperatures u1/u1exact along the circular arc between AD and BC of two-dimensional domain associated with quarter of ring
Fig.5  Relative errors of SBFE solutions versus number of degrees of freedom used in discretization of circular defining curve. DF: Defining curve
Fig.6  Schematics of (a) plane-strain hollowed disk fully restrained against the movement on its outer boundary and subjected to uniform shear traction on its inner boundary and (b) equivalent quarter of body used in analysis
n u1/u1exact
θ/π=0.1 θ/π=0.2 θ/π=0.3 θ/π=0.4
Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2
2 0.7449 0.9529 0.7832 1.0019 0.7702 0.9853 0.7554 0.9664
4 0.9411 1.0009 0.9330 0.9923 0.9320 0.9913 0.9379 0.9975
8 0.9829 0.9981 0.9840 0.9993 0.9843 0.9996 0.9826 0.9979
16 0.9960 0.9998 0.9956 0.9995 0.9956 0.9995 0.9960 0.9999
Tab.2  Normalized displacements u1/u1exact along circular arc between AD and BC of plane-strain hollowed disk
n u2/u2exact
θ/π=0.1 θ/π=0.2 θ/π=0.3 θ/π=0.4
Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2
2 0.7554 0.9664 0.7702 0.9853 0.7832 1.0019 0.7449 0.9529
4 0.9379 0.9975 0.9320 0.9913 0.9330 0.9923 0.9411 1.0009
8 0.9826 0.9979 0.9843 0.9996 0.9840 0.9993 0.9829 0.9981
16 0.9960 0.9999 0.9956 0.9995 0.9956 0.9995 0.9960 0.9998
Tab.3  Normalized displacements u2/u2exact along circular arc between AD and BC of plane-strain hollowed disk
Fig.7  Relative errors of SBFE solutions versus number of degrees of freedom used in discretization of circular defining curve. DF: Defining curve
n u1/u1exact
x1=1.25 x1=1.50 x1=1.75 x1=2.00
Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2
2 0.7368 0.8935 0.7228 0.9262 0.7799 1.0043 0.8605 1.1004
4 0.9361 0.9806 0.9309 0.9936 0.9385 1.0115 0.9413 1.0199
8 0.9830 0.9946 0.9812 0.9974 0.9830 1.0014 0.9852 1.0046
16 0.9957 0.9986 0.9952 0.9993 0.9957 1.0003 0.9963 1.0011
Tab.4  Normalized displacements u1/u1exact along the line x1=x2 of quarter of hollowed circular plate
n u2/u2exact
x1=1.25 x1=1.50 x1=1.75 x1=2.00
Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2
2 1.0289 0.9705 1.0421 0.9801 1.0493 0.9917 1.0525 0.9981
4 1.0072 0.9984 1.0088 0.9987 1.0097 0.9998 1.0111 1.0019
8 1.0016 0.9997 1.0021 0.9998 1.0023 0.9999 1.0022 1.0000
16 1.0004 0.9999 1.0005 1.0000 1.0006 1.0000 1.0005 1.0000
Tab.5  Normalized displacements u2/u2exact along the line x1=x2 of quarter of hollowed circular plate
n u3/u3exact
x1=1.25 x1=1.50 x1=1.75 x1=2.00
Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2
2 0.9864 1.0111 0.9699 1.0088 0.9500 0.9992 0.9268 0.9844
4 0.9973 1.0018 0.9944 1.0019 0.9914 1.0014 0.9892 1.0010
8 0.9993 1.0004 0.9986 1.0004 0.9978 1.0003 0.9970 1.0000
16 0.9998 1.0001 0.9996 1.0001 0.9994 1.0001 0.9992 1.0000
Tab.6  Normalized electric potential u3/u3exact along the line x1=x2 of quarter of hollowed circular plate
Fig.8  Relative errors of SBFE solutions versus number of degrees of freedom used in discretization of circular defining curve. DF: Defining curve
1 J PWolf. The Scaled Boundary Finite Element Method. Chichester: John Wiley & Sons, 2003
2 J PWolf, C Song. Finite-Element Modelling of Unbounded Domain. Chichester: John Wiley & Sons, 1996
3 J ADeeks, J P Wolf. A virtual work derivation of the scaled boundary finite-element method for elastostatics. Computational Mechanics, 2002, 28(6): 489–504
https://doi.org/10.1007/s00466-002-0314-2
4 T ACruse. Boundary Element Analysis in Computational Fracture Mechanics. Dordrecht: Kluwer Academic Publishers, 1988
5 C ABrebbia, J Dominguez. Boundary Elements: An Introductory Course. 2nd ed. New York: McGraw-Hill, 1989
6 MBonnet, G Maier, CPolizzotto. Symmetric Galerkin Boundary Element Methods. Applied Mechanics Reviews, 1998, 51(11): 669–703
https://doi.org/10.1115/1.3098983
7 JLiu, G A Lin. A scaled boundary finite element method applied to electrostatic problems. Engineering Analysis with Boundary Elements, 2012, 36(12): 1721–1732
https://doi.org/10.1016/j.enganabound.2012.06.010
8 CLi, H Man, CSong, WGao. Fracture analysis of piezoelectric materials using the scaled boundary finite element method. Engineering Fracture Mechanics, 2013, 97: 52–71
https://doi.org/10.1016/j.engfracmech.2012.10.019
9 T HVu, A J Deeks. Using fundamental solutions in the scaled boundary finite element method to solve problems with concentrated loads. Computational Mechanics, 2014, 53(4): 641–657
https://doi.org/10.1007/s00466-013-0923-y
10 E TOoi, C Song, FTin-Loi. A scaled boundary polygon formulation for elasto-plastic analyses. Computer Methods in Applied Mechanics and Engineering, 2005, 268: 905–937
https://doi.org/10.1016/j.cma.2013.10.021
11 J PDoherty, A J Deeks. Adaptive coupling of the finite-element and scaled boundary finite-element methods for non-linear analysis of unbounded media. Computers and Geotechnics, 2015, 32(6): 436–444
https://doi.org/10.1016/j.compgeo.2005.07.001
12 FLi, P Ren. A novel solution for heat conduction problems by extending scaled boundary finite element method. International Journal of Heat and Mass Transfer, 2016, 95: 678–688
https://doi.org/10.1016/j.ijheatmasstransfer.2015.12.019
13 MLi, H Zhang, HGuan. Study of offshore monopole behavior due to ocean waves. Ocean Engineering, 2011, 38(17–18): 1946–1956
https://doi.org/10.1016/j.oceaneng.2011.09.022
14 X NMeng, Z J Zou. Radiation and diffraction of water waves by an infinite horizontal structure with a sidewall using SBFEM. Ocean Engineering, 2013, 60: 193–199
https://doi.org/10.1016/j.oceaneng.2012.12.017
15 HGravenkamp, C Birk, CSong. The computation of dispersion relations for axisymmetric waveguides using the scaled boundary finite element method. Ultrasonics, 2014, 54(5): 1373–1385
https://doi.org/10.1016/j.ultras.2014.02.004
16 CLi, E T Ooi, C Song, SNatarajan. SBFEM for fracture analysis of piezoelectric composites under thermal load. International Journal of Solids and Structures, 2015, 52: 114–129
https://doi.org/10.1016/j.ijsolstr.2014.09.020
17 CSong, J P Wolf. The scaled boundary finite-element method—alias consistent infinitesimal finite-element cell method—for elastodynamics. Computer Methods in Applied Mechanics and Engineering, 1997, 147(3–4): 329–355
https://doi.org/10.1016/S0045-7825(97)00021-2
18 J PWolf, C Song. The scaled boundary finite-element method: A fundamental solution-less boundary-element method. Computer Methods in Applied Mechanics and Engineering, 2001, 190(42): 5551–5568
https://doi.org/10.1016/S0045-7825(01)00183-9
19 A JDeeks. Prescribed side-face displacements in the scaled boundary finite-element method. Computers & Structures, 2004, 82(15–16): 1153–1165
https://doi.org/10.1016/j.compstruc.2004.03.024
20 CSong, J P Wolf. Body loads in scaled boundary finite-element method. Computer Methods in Applied Mechanics and Engineering, 1999, 180(1–2): 117–135
https://doi.org/10.1016/S0045-7825(99)00052-3
21 YHe, H Yang, A JDeeks. An element-free Galerkin (EFG) scaled boundary method. Finite Elements in Analysis and Design, 2012, 62: 28–36
https://doi.org/10.1016/j.finel.2012.07.001
22 T HVu, A J Deeks. Use of higher-order shape functions in the scaled boundary finite element method. International Journal for Numerical Methods in Engineering, 2006, 65(10): 1714–1733
https://doi.org/10.1002/nme.1517
23 YHe, H Yang, A JDeeks. Use of Fourier shape functions in the scaled boundary method. Engineering Analysis with Boundary Elements, 2014, 41: 152–159
https://doi.org/10.1016/j.enganabound.2014.01.012
24 A JDeeks, J P Wolf. An h-hierarchical adaptive procedure for the scaled boundary finite-element method. International Journal for Numerical Methods in Engineering, 2002, 54(4): 585–605
https://doi.org/10.1002/nme.440
25 T HVu, A J Deeks. A p-adaptive scaled boundary finite element method based on maximization of the error decrease rate. Computational Mechanics, 2007, 41(3): 441–455
https://doi.org/10.1007/s00466-007-0203-9
26 A JDeeks, C E Augarde. A meshless local Petrov-Galerkin scaled boundary method. Computational Mechanics, 2005, 36(3): 159–170
https://doi.org/10.1007/s00466-004-0649-y
27 N VChung. Analysis of two-dimensional linear field problems by scaled boundary finite element method. Dissertation for the Doctoral Degree. Bangkok: Chulalongkorn University, 2016
28 N VChung, R Jaroon, PPhoonsak. Scaled boundary finite element method for two-dimensional linear multi-field media. Engineering Journal (Thailand), 2017, 21(7): 334–360
29 E TOoi, C Song, FTin-Loi, Z JYang. Automatic modelling of cohesive crack propagation in concrete using polygon scaled boundary finite elements. Engineering Fracture Mechanics, 2012, 93: 13–33
https://doi.org/10.1016/j.engfracmech.2012.06.003
30 E TOoi, C Shi, CSong, FTin-Loi, Z JYang. Dynamic crack propagation simulation with scaled boundary polygon elements and automatic remeshing technique. Engineering Fracture Mechanics, 2013, 106: 1–21
https://doi.org/10.1016/j.engfracmech.2013.02.002
31 RDieringer, W Becker. A new scaled boundary finite element formulation for the computation of singularity orders at cracks and notches in arbitrarily laminated composites. Composite Structures, 2015, 123: 263–270
https://doi.org/10.1016/j.compstruct.2014.12.036
32 SNatarajan, J Wang, CSong, CBirk. Isogeometric analysis enhanced by the scaled boundary finite element method. Computer Methods in Applied Mechanics and Engineering, 2015, 283: 733–762
https://doi.org/10.1016/j.cma.2014.09.003
33 B HNguyen, H D Tran, C Anitescu, XZhuang, TRabczuk. Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems. Computer Methods in Applied Mechanics and Engineering, 2017, 323: 132–150
https://doi.org/10.1016/j.cma.2017.05.011
34 B HNguyen, H D Tran, C Anitescu, XZhuang, TRabczuk. An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems. Computer Methods in Applied Mechanics and Engineering, 2016, 306: 252–275
https://doi.org/10.1016/j.cma.2016.04.002
35 PLi, J Liu, GLin, PZhang, BXu. A combination of isogeometric technique and scaled boundary method for solution of the steady-state heat transfer problems in arbitrary plane domain with Robin boundary. Engineering Analysis with Boundary Elements, 2017, 82: 43–56
https://doi.org/10.1016/j.enganabound.2017.05.006
36 FLi, T Qiang. The scaled boundary finite element analysis of seepage problems in multi-material regions. International Journal of Computational Methods, 2012, 9(1): 1240008
https://doi.org/10.1142/S0219876212400087
Related articles from Frontiers Journals
[1] Chung Nguyen VAN. Numerical investigation of circle defining curve for two-dimensional problem with general boundaries using the scaled boundary finite element method[J]. Front. Struct. Civ. Eng., 2019, 13(1): 92-102.
[2] DU Jianguo, LIN Gao. Improved numerical method for time domain dynamic structure-foundation interaction analysis based on scaled boundary finite element method[J]. Front. Struct. Civ. Eng., 2008, 2(4): 336-342.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed