A. Kamitani

A. Saitoh

H. Nakamura

S. Ikuno

S. Nakata

T. Itoh

T. Takayama

Y. Tamura

National Institute for Fusion Science (NIFS-Repository)

Introduction Before executing a numerical code based onthe Finite Element Method (FEM) or the Boundary ElementMethod (BEM), a region must be divided into a set of ele-ments. For the purpose of removing the element decompo-sition from a numerical code, meshless approaches such theFinite Node Method (FNM)1, 2, 3, 4) and the Boundary NodeMethod (BNM)5, 6, 7) have been so far proposed. However,they have been plagued by various diﬃculties. First of all,in the FNM, the Lagrange multiplier method is adopted asthe implementation method1, 3, 4) of essential boundary con-ditions. Thus, the boundary value problem of the governingequation must be first rewritten into the equivalent variationalform. On the other hand, the number of unknowns for theBNM is twice as many as that for the BEM5, 6, 7) and, hence,the computational cost required for the solution of the dis-cretized equation in the BNM becomes about 8 times as muchas that in the BEM.The purpose of the present study is to algebraically for-mulate the FNM without using the variational principle. Inaddition, we propose the method for reducing the number ofunknowns for the BNM by half.eXtended FNM For simplicity, we consider the followingtwo-dimensional (2D) Poisson problem on the domain Ωbounded by a simple closed curve ∂Ω:−∇2u = p in Ω, (1)u = u¯ on ΓD, (2)∂u∂n= q¯ on ΓN, (3)where ΓD ∪ ΓN = ∂Ω and ΓD ∩ ΓN = ϕ. In addition, n denotesan outward unit normal on ∂Ω. Furthermore, the superposedbar indicates prescribed boundary values and p(x) is a givenfunction on Ω. As is well known, both eqs. (1) and (3) aresatisfied if and only if the following weak form is fulfilled:∀w s.t. w|ΓD = 0 :Ω∇w · ∇u d2x =Ωwp d2x +∫ΓNwq¯ ds = 0. (4)Let both the test function w(x) and the trial function u(x)satisfy w, u ∈ span(ϕ1, ϕ2, · · · , ϕN), and let the function spaceon ΓD be limited to span(N1,N2, · · · ,NK). Then, as the dis-cretized form of the weak form (4), the following propositionis obtained :∀w ∈ KerCT : wT (Au − f ) = 0, (5)where N-dimensional vectors, w and u, correspond w(x) andu(x), respectively. In addition, A, C and f denote an N × Nmatrix, an N×K matrix and an N-dimensional vector, respec-tively. By taking account of KerCT = (ImC)⊥, the aboveproposition (5) is equivalent to Au− f ∈ ImC. Moreover, thediscretized form of eq. (2) can be written asCTu = g. Hence,the discretized form of the 2D Poisson problem can be writtenin the form, [A CCT O] [uλ]=[fg]. (6)Note that eq. (6) coincides with the matrix equation for theFNM. In this way, the FNM is algebraically formulated with-out employing the variational principle.Although the 2D Poisson problem is used to demon-strate the algebraic formulation of the FNM, the discretizationprocedure used in the present study is applicable to generalboundary-value problems.eXtended BNM As mentioned above, the number of un-knowns for the BNM is twice as many as that for the BEM.The reason for this is explained as follows. In the conven-tional BNM, the shape functions {ψi(s)}Ni=1 are determinedby means of the Moving-Least Square (MLS) approximationand, therefore, the functions do not at all satisfy the deltafunction property: ψi(s j) = δi j. However, it is the deltafunction property that plays an important role in reducing thenumber of unknowns in the discretized equation. In contrast,the property is always fulfilled by the shape functions of theBEM. Hence, the number of unknowns for the BEM is equalto the number of nodes on the boundary.In order to have the delta function property satisfied bythe shape functions, we employ not the MLS approximation1)but the Radial Point Interpolation Method (RPIM)8) to the de-termination of the shape functions. As a result, the CPU timerequired for the solution of the discretized equation success-fully reduces by a factor of eight.From the above result, we can conclude that the RPIMis useful to accelerate the BNM.1) Belytschko, T. et al.: Int. J. Numer. Meth. Engng. 37(1994) 229.2) Atluri, S. N. and Zhu, T.: Comput. Mech. 22 (1998) 117.3) Ikuno, S. et al.: IEEE Trans. Magn. 43 (2007) 1501.4) Kamitani, A. et al.: Plasma Fusion Res. 6 (2011) 2401074.5) Mukherjee, Y. X. and Mukherjee, S.: Int. J. Numer. Meth.Engng. 40 (1997) 797.6) Chati, M. K. et al.: Int. J. Numer. Meth. Engng. 50 (2001)2233.7) Saitoh A. et al.: IEEE Trans. Magn. 47 (2011) 1222.8) Wang, J. G. and Liu, G. R.: Int. J. Numer. Meth. Engng.54 (2002) 1623.444§41. Development of Extended Finite/Boundary Node MethodKamitani, A., Takayama, T. (Yamagata Univ.),  Ikuno, S., Itoh, T. (Tokyo Univ. of Technology),  Nakata, S. (Ritsumeikan Univ.),  Saitoh, A. (Univ. Hyogo),  Tamura, Y. (Konan Univ.),  Nakamura, H.

Development of Extended Finite/ Boundary Node Method

Development of Extended Finite/ Boundary Node Method

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