A new scaled boundary finite element method using Fourier shape. functions

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1 APCOM & ISCM -4 th December, 3, Singapore A new caled boundary finite element method uing Fourier hape Abtract function Yiqian He¹, Haitian Yang¹, and *Andrew J. Deek Department of Engineering Mechanic, Dalian Univerity of Technology, Dalian 64, P.R. China School of Engineering and Computing Science, Durham Univerity, South Road, Durham, DH 3LE, UK *Correponding author: a.j.deek@durham.ac.uk The caled boundary finite element method (SBFEM) i a emi-analytical method, whoe veratility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problem. Thi paper invetigate the poibility of uing Fourier hape function in the SBFEM to form the approximation in the circumferential direction. The hape function effectively form a Fourier erie expanion in the circumferential direction, and are augmented by additional linear hape function. The propoed method i evaluated by olving elatotatic problem. The accuracy and convergence of the propoed method i demontrated, and the performance i found to be better than uing polynomial element or uing an element-free Galerkin approximation for the circumferential approximation. Keyword: Scaled boundary method; Fourier hape function; computational accuracy; tre ingularitie; unbounded domain Introduction The caled boundary method (SBM) i a emi-analytical method developed relatively recently by Wolf and Song (Wolf and Song, 996). The method introduce a normalied radial coordinate ytem baed on a caling centre and a defining curve (uually taken a the boundary). The governing differential equation are weakened in the circumferential direction and then olved analytically in the normalied radial direction. The SBM combine the advantage of the Finite Element Method (FEM) and the Boundary Element Method (BEM), and, unlike the BEM, no fundamental olution i required. In addition, the SBM ha been hown to be more efficient than the FEM for problem involving unbounded domain and for problem involving tre ingularitie or dicontinuitie (Deek and Wolf, ). Effective application of thi method have been demontrated in variou problem domain, including fracture problem and foundation problem. In the caled boundary method, the dicretiation approach ued in the circumferential direction ha ignificant influence on the accuracy of the reulting olution (Deek and Augarde, 5). The mot commonly ued method for performing thi circumferential dicretiation i the finite element approach, leading to the method called the caled boundary finite element method (SBFEM). Vu and Deek (Vu and Deek, 6, 8a, 8b) invetigated the ue of higher-order polynomial hape function in the SBFEM, and demontrated the SBFEM converged

2 ignificantly fater under p-refinement than under h-refinement. The development of mehle method provided another approach to building circumferential approximation for the caled boundary method. Deek and Augarde (Deek and Augarde, 5) developed a Mehle Local Petrov-Galerkin method caled boundary method (MLPG-SBM) and and He et al (He et al, ) developed an Element-free Galerkin caled boundary method (EFG-SBM). Thi work howed that thee two mehle caled boundary method gave a higher level of accuracy and rate of convergence than the conventional SBFEM uing linear or quadratic element, with the EFG-SBM performing lightly better than the MLPG-SBM. In thi paper, the poibility of uing hape function baed on the term of a Fourier erie for the circumferential approximation of the SBFEM i invetigated. Fourier interpolation containing trigonometric function have been applied to both the finite element method (FEM) and the boundary element method (BEM). For example, Guan et al. (Guan et al, 6) developed a Fourier erie baed FEM for the analyi of tube hydroforming, and howed that thi Fourier hape function reduced the number of degree of freedom required. Javaran and Khaji (Javaran, ; Khaji and Javaran, 3) applied Fourier radial bai function into the BEM, and howed that the reulting BEM i much more accurate than the BEM uing claic Lagrange hape function. Although the advantage of Fourier baed FEM and BEM have been illutrated in previou work, to date there ha been no work reported on the ue of Fourier hape function in the SBFEM. A new Fourier-baed caled boundary method (F-SBM) i preented in thi paper. A et of hape function baed on Fourier erie expanion i derived, and augmented with linear hape function. The new hape function provide good approximation to both trigonometric and polynomial function in the circumferential direction of the caled boundary ytem. In the numerical example, the F-SBM i ued to olve a two-dimenional elatotatic problem. The accuracy and convergence of F-SBM i compared with the conventional SBFEM uing both linear and quadratic element and with the EFG-SBM. Superior performance in term of both accuracy and convergence i demontrated. A Fourier hape function Thi paper employ hape function obtained from the well-known Fourier erie. Baed on the theory of the Fourier erie, any continuou function f() r maybe repreented by a erie of trigonometric function a nπ nπ f( r) = a + anco r + bnin r n= Lmax L () max where a, a n, b n and L max repreent the Fourier erie parameter. Thu on the boundary at ξ =, the diplacement can be approximated a m jπ jπ uh( ) = a + aj co + bjin j= L L () where i the circumferential coordinate in caled boundary element, L i the length of the boundary and m repreent the order of Fourier erie. To preerve C continuity between the edge or element, linear polynomial function term are added into the tandard Fourier approximation a m L jπ jπ uh( ) = α + β + aj co bjin L L + j= L L (3)

3 where α and β repreent the value of the function at the end node of the element. While it i poible to ue the Fourier parameter a the unknown boundary parameter when olving the caled boundary finite element equation, here the parameter in the Fourier expanion above are tranferred to nodal value at equally paced node along each element for eae of applying eential boundary condition m + node are ued, the nodal and enforcing C continuity between element. If ( ) value vector { u } can be related to the parameter in Equation (3) by { u} [ T] { uˆ } = (4) T m m where { uˆ } { α a a b b β } =, and [ T ] i a tranfer matrix aembled a Tij = ψ j ( Si ) i, j =, m+ (5) where S i i the circumferential coordinate of the i th node, and the component function of the Fourier expanion are L i = L ( i ) π co i m+ L ψ i () = (6) ( i m) π in m + i m + L i= m+ L Inverting Equation (4), the parameter { û } in Equation (3) can be related to the nodal value {u h } by { uˆ } = [ T] { u} (7) Thu the approximation for diplacement can be rewritten a () { } [ ] uh = ψ T { u} (8) The hape function relating to the nodal diplacement are hence { ϕ} = { ψ} [ T ] (9) and the hape function matrix for the caled boundary method then become ϕ( )... ϕm+ ( ) [ N ( )] = ϕ( )... ϕm+ ( ) () Figure 3 plot thee Fourier hape function for m =, where 6 node are required. 3

4 .5 ϕ ϕ ϕ 3 ϕ 4 ϕ 5 ϕ 6 ϕ() Figure The Fourier hape function for order m = Performance of the method An infinite plate with a through crack The example refer to the problem of determining the mode I tre intenity factor (SIF) K for a through crack in an infinite plate, a illutrated in Figure 9. The I applied tre σ =. Due to the ymmetry, one quarter of the problem i modelled, a hown in Figure 3, with the model coniting of a bounded domain, with the caling centre at the crack tip (point E), and an unbounded domain, with the caling centre at the middle of crack (point A). The node are introduced on the edge AB, BC and CD with uniform pacing, d. The problem ha an exact olution, = σ πa. KI Figure Infinite plate with a through crack: geometry and load 4

5 Figure 3 Scaled boundary model of an infinite plate with a through crack. In Table the F-SBM olution are compared with the SBFEM with linear element and the EFG-SBM with linear bai. The reult how that the F-SBM achieve high accuracy for SIF, for example, a relative error a low a.555% can be obtained uing 53 node. In comparion with SBFEM and EFG-SBM, it can be een that F-SBM ha higher accuracy when the ame number of node are ued. Table The reult of SIF uing F-SBM Number of node F-SBM Error% SBFEM (Linear) Error% EFG-SBM (Linear) Error% Exact Solution e e e e e e e e e e e e e e e e e e-3 5

6 Concluion A new SBFEM uing Fourier hape function i preented in thi paper. The hape function are baed on the Fourier erie expanion and augmented with additional linear hape function term. By uing a tranfer matrix, the nodal value are related with Fourier parameter, and in thi way the eential boundary condition can be conveniently handled. In the numerical example, the new approach ha been hown to yield higher accuracy and fater convergence in comparion with the SBFEM uing linear or quadratic element and the EFG-SBM uing linear or quadratic bai. Reference Wolf, J. P and Song, Ch (996), Finite-Element Modelling of Unbounded Media. John Wiley and Son: Chicheter. Deek, A. J. and Wolf, J. P. (a), A virtual work derivation of the caled boundary finite-element method for elatotatic, Comput. Mech. 8, pp Deek, A. J. and Augarde, C. E. (5), A mehle local Petrov-Galerkin caled boundary method. Comput. Mech. 36, pp Guan, Y., Pourboghrat F. and Yu, W. R. (6), Fourier erie baed finite element analyi of tube hydroforming: A axiymmetric model. Engineering Computation 3(7), pp He, Y. Q., Yang, H. T. and Deek, A. J. (), An Element-free Galerkin (EFG) caled boundary method, Fin. Elem. Analy. De. 6, pp Javaran, H. S., Khaji N. and Moharrami H. (), A dual reciprocity BEM approach uing new Fourier radial bai function applied to D elatodynamic tranient analyi. Engineering Analyi with Boundary Element 35, pp Khaji N. and Javaran, H. S. (3), New complex Fourier hape function for the analyi of twodimenional potential problem uing boundary element method. Engineering Analyi with Boundary Element 37, pp Vu, T. H. and Deek, A. J. (6), Ue of higher order hape function in the caled boundary finite element method, Int. J. Numer. Meth. Eng. 65, pp Vu, T. H. and Deek, A. J. (8a), A p-hierarchical adaptive procedure for the caled boundary finite element method, Int. J. Numer. Meth. Eng. 73, pp Vu, T. H. and Deek, A. J. (8b), A p-adaptive caled boundary finite element method baed on maximization of the error decreae rate, Comput. Mech. 4, pp

CRACK TIP STRESS FIELDS FOR ANISOTROPIC MATERIALS WITH CUBIC SYMMETRY

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