Solving Elasto-Static Bounded Problems with a Novel Arbitrary-Shaped Element
Abstract
A simple method to analysis any arbitrary domain shapes with a single element which based on Decoupled Scaled Boundary Finite Element Method is presented in this paper. The introduced element is based on boundary finite element method which helps to modelling curve and sharp boundaries with acceptable accuracy. Shape functions and mapping functions are similar to Decoupled Scaled Boundary Finite Element Method but locating center origin (LCO) is relocated in this method from corners with direct view to whole domain into shape center and formulation and behavior of the method is developed for the element. The most important advantageous of this technique is ability of solving displacement in domain by solving differential equations which causes more accurate answers in domain. We also perform well-established numerical tests and show the performance of the new element. Results shown us the accuracy and reliable answers for the introduced element. Also some benchmark examples are solved by this method and answers are compared with correct answers and plotted. High accuracy of answers with low cost of calculations and ability of the method to analysis the curve and sharp boundaries are the most important advantageous of this new element.
Keywords
References
Bathe, Klaus-Jürgen, Finite element procedures. Klaus-Jurgen Bathe, (2017).
Brebbia, C. A., J. Dominguez, and J. L. Tassoulas. “Boundary Elements: An Introductory Course.” Journal of Applied Mechanics 58, no. 3 (1991): 860. doi:10.1115/1.2897280.
Wolf, J.P., and C. Song. “The Scaled Boundary Finite-Element Method - A Primer: Derivations.” Advances in Computational Structural Mechanics (n.d.). doi:10.4203/ccp.55.2.3.
Hell, Sascha, and Wilfried Becker. “An Enriched Scaled Boundary Finite Element Method for 3D Cracks.” Engineering Fracture Mechanics 215 (June 2019): 272–293. doi:10.1016/j.engfracmech.2019.04.032.
Song, Chongmin, Ean Tat Ooi, and Sundararajan Natarajan. “A Review of the Scaled Boundary Finite Element Method for Two-Dimensional Linear Elastic Fracture Mechanics.” Engineering Fracture Mechanics 187 (January 2018): 45–73. doi:10.1016/j.engfracmech.2017.10.016.
Strang, Gilbert, and George J. Fix. An analysis of the finite element method. Vol. 212. Englewood Cliffs, NJ: Prentice-hall, 1973.
Zienkiewicz, O.C., R.L. Taylor, and J.Z. Zhu. “Adaptive Finite Element Refinement.” The Finite Element Method Set (2005): 500–524. doi:10.1016/b978-075066431-8.50182-x.
Brezzi, Franco, Jim Jr. Douglas, Michel Fortin, and L. Donatella Marini. “Efficient Rectangular Mixed Finite Elements in Two and Three Space Variables.” ESAIM: Mathematical Modelling and Numerical Analysis 21, no. 4 (1987): 581–604. doi:10.1051/m2an/1987210405811.
Arnold, Douglas N., and Ragnar Winther. “Mixed Finite Elements for Elasticity in the Stress-Displacement Formulation.” Current Trends in Scientific Computing (2003): 33–42. doi:10.1090/conm/329/05839.
Arnold, Douglas N., Richard S. Falk, and Ragnar Winther. “Finite Element Exterior Calculus, Homological Techniques, and Applications.” Acta Numerica 15 (May 2006): 1–155. doi:10.1017/s0962492906210018.
Komatitsch, Dimitri, and Jeroen Tromp. “Spectral-Element Simulations of Global Seismic Wave Propagation-I. Validation.” Geophysical Journal International 149, no. 2 (May 2002): 390–412. doi:10.1046/j.1365-246x.2002.01653.x.
Davis, Christopher, June G. Kim, Hae-Soo Oh, and Min Hyung Cho. “Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation.” Journal of Scientific Computing 55, no. 1 (September 25, 2012): 200–230. doi:10.1007/s10915-012-9645-0.
Hall, W. S. “Boundary Element Method.” Solid Mechanics and Its Applications (1994): 61–83. doi:10.1007/978-94-011-0784-6_3.
Liu, Lei, Junqi Zhang, Chongmin Song, Carolin Birk, and Wei Gao. “An Automatic Approach for the Acoustic Analysis of Three-Dimensional Bounded and Unbounded Domains by Scaled Boundary Finite Element Method.” International Journal of Mechanical Sciences 151 (February 2019): 563–581. doi:10.1016/j.ijmecsci.2018.12.018.
Natarajan, Sundararajan, Ean Tat Ooi, Albert Saputra, and Chongmin Song. “A Scaled Boundary Finite Element Formulation over Arbitrary Faceted Star Convex Polyhedra.” Engineering Analysis with Boundary Elements 80 (July 2017): 218–229. doi:10.1016/j.enganabound.2017.03.007.
Liu, Lei, Junqi Zhang, Chongmin Song, Carolin Birk, Albert A. Saputra, and Wei Gao. “Automatic Three-Dimensional Acoustic-Structure Interaction Analysis Using the Scaled Boundary Finite Element Method.” Journal of Computational Physics 395 (October 2019): 432–460. doi:10.1016/j.jcp.2019.06.033.
Atluri, S. N., and T. Zhu. “A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics.” Computational Mechanics 22, no. 2 (August 24, 1998): 117–127. doi:10.1007/s004660050346.
Yumin, Cheng, and Peng Miaojuan. “Boundary Element-Free Method for Elastodynamics.” Science in China Series G: Physics, Mechanics and Astronomy 48, no. 2 (March 2005): 641–657. doi:10.1007/bf02687411.
LIU, G.R., and Y.T. GU. “A Local Radial Point Interpolation Method (Lrpim) For Free Vibration Analyses Of 2-D Solids.” Journal of Sound and Vibration 246, no. 1 (September 2001): 29–46. doi:10.1006/jsvi.2000.3626..
Zhu, T., J.-D. Zhang, and S. N. Atluri. “A Local Boundary Integral Equation (LBIE) Method in Computational Mechanics, and a Meshless Discretization Approach.” Computational Mechanics 21, no. 3 (April 23, 1998): 223–235. doi:10.1007/s004660050297.
Babuška, Ivo, Fabio Nobile, and Raúl Tempone. “A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data.” SIAM Journal on Numerical Analysis 45, no. 3 (January 2007): 1005–1034. doi:10.1137/050645142.
Roberts, J.E., and J.-M. Thomas. “Mixed and Hybrid Methods.” Handbook of Numerical Analysis (1991): 523–639. doi:10.1016/s1570-8659(05)80041-9..
Chen, Linchong, and Xiaolin Li. “Boundary Element-Free Methods for Exterior Acoustic Problems with Arbitrary and High Wavenumbers.” Applied Mathematical Modelling 72 (August 2019): 85–103. doi:10.1016/j.apm.2019.03.017.
Chen, L., X. Liu, and X. Li, The boundary element-free method for 2D interior and exterior Helmholtz problems. Computers & Mathematics with Applications, 2019. 77(3): p. 846-864. doi: j.camwa.2018.10.022.
Bu, Weiping, Yifa Tang, Yingchuan Wu, and Jiye Yang. “Finite Difference/finite Element Method for Two-Dimensional Space and Time Fractional Bloch–Torrey Equations.” Journal of Computational Physics 293 (July 2015): 264–279. doi:10.1016/j.jcp.2014.06.031.
Khodakarami, M.I. and H.K. Hedayati, DSBFEM AS A MACRO ELEMENT FOR FE ANALYSIS, in the 12th word congress on computional mechanics. 2016: Korea, Seoul.
Khodakarami, M.I., N. Khaji, and M.T. Ahmadi. “Modeling Transient Elastodynamic Problems Using a Novel Semi-Analytical Method Yielding Decoupled Partial Differential Equations.” Computer Methods in Applied Mechanics and Engineering 213–216 (March 2012): 183–195. doi:10.1016/j.cma.2011.11.016..
Khodakarami, M.I., and N. Khaji. “Analysis of Elastostatic Problems Using a Semi-Analytical Method with Diagonal Coefficient Matrices.” Engineering Analysis with Boundary Elements 35, no. 12 (December 2011): 1288–1296. doi:10.1016/j.enganabound.2011.06.003..
Khodakarami, M.I., and N. Khaji. “Analysis of Elastostatic Problems Using a Semi-Analytical Method with Diagonal Coefficient Matrices.” Engineering Analysis with Boundary Elements 35, no. 12 (December 2011): 1288–1296. doi:10.1016/j.enganabound.2011.06.003..
Khaji, N., and M.I. Khodakarami. “A New Semi-Analytical Method with Diagonal Coefficient Matrices for Potential Problems.” Engineering Analysis with Boundary Elements 35, no. 6 (June 2011): 845–854. doi:10.1016/j.enganabound.2011.01.011..
Chandrupatla, Tirupathi R., Ashok D. Belegundu, T. Ramesh, and Chaitali Ray. Introduction to finite elements in engineering. Vol. 10. Upper Saddle River, NJ: Prentice Hall, 2002..
Miers, L.S., and J.C.F. Telles. “Meshless Boundary Integral Equations with Equilibrium Satisfaction.” Engineering Analysis with Boundary Elements 34, no. 3 (March 2010): 259–263. doi:10.1016/j.enganabound.2009.09.008.
DOI: 10.28991/cej-2019-03091384
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