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Recent Advances and Emerging Applications of the Boundary Element Method Y Recent Advances and Emerging Applications of the Boundary Element Method Y. J. Liu, S. Mukherjee, N. Nishimura, M. Schanz, W. Ye, A. Sutradhar, E. Pan, A. Dumont, A. Frangi, A. Saez To cite this version: Y. J. Liu, S. Mukherjee, N. Nishimura, M. Schanz, W. Ye, et al.. Recent Advances and Emerging Applications of the Boundary Element Method. Applied Mechanics Reviews, American Society of Mechanical Engineers, 2011, 64 (3), 10.1115/1.4005491. hal-01401752 HAL Id: hal-01401752 https://hal.archives-ouvertes.fr/hal-01401752 Submitted on 23 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License 3FDFOU "EWBODFT BOE &NFSHJOH "QQMJDBUJPOT PG UIF #PVOEBSZ &MFNFOU .FUIPE 4QPOTPSFE CZ UIF 64 /BUJPOBM 4DJFODF 'PVOEBUJPO B XPSLTIPQ PO UIF CPVOEBSZ FMFNFOU NFUIPE #&. XBT IFME PO UIF DBNQVT PG UIF 6OJWFSTJUZ PG "LSPO EVSJOH 4FQUFNCFS o /4' i8PSLTIPQ PO UIF &NFSHJOH "QQMJDBUJPOT BOE 'VUVSF %JSFDUJPOT PG UIF #PVOEBSZ &MFNFOU .FUIPE w 6OJWFSTJUZ PG "LSPO 0IJP 4FQUFNCFS o 5IJT QBQFS XBT QSFQBSFE BGUFS UIJT XPSLTIPQ CZ UIF PSHBOJ[FST BOE QBSUJDJQBOUT CBTFE PO UIF QSFTFOUBUJPOT BOE EJTDVTTJPOT BU UIF XPSLTIPQ 5IF QBQFS BJNT UP SFWJFX UIF NBKPS SFTFBSDI BDIJFWFNFOUT JO UIF MBTU EFDBEF UIF DVSSFOU TUBUVT BOE UIF GVUVSF EJSFDUJPOT PG UIF #&. 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BOE SFMBUFE BQQSPBDIFT EJTDVTTFE JO UIF SFWJFX DBO CF GPVOE JO UIF 3FGFSFODF TFDUJPO XJUI NPSF UIBO QBQFST DJUFE JO UIJT SFWJFX Y. J. Liu Mechanical Engineering, University of Cincinnati, 598 Rhodes Hall, 2600 Clifton Avenue, Cincinnati, OH 45221-0072 e-mail: [email protected] S. Mukherjee Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853 e-mail: [email protected] N. Nishimura Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan e-mail: [email protected] M. Schanz Institute of Applied Mechanics, Graz University of Technology, Technikerstr. 4, A-8010 Graz, Austria e-mail: [email protected] W. Ye Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong, China e-mail: [email protected] A. Sutradhar Department of Surgery and Department of Mechanical Engineering, The Ohio State University, 915 Olentangy River Road, Suite 2100, Columbus, OH 43212 e-mail: [email protected] E. Pan Department of Civil Engineering, The University of Akron, ASEC Room 534, Akron, OH 44325-3905 e-mail: [email protected] N. A. Dumont Pontifical Catholic University of Rio de Janeiro, 22451-900 Brazil e-mail: [email protected] A. Frangi Department of Structural Engineering, Politecnico of Milano, P.za L. da Vinci 32, 20133 Milano, Italy e-mail: [email protected] A. Saez Department of Continuum Mechanics and Structural Analysis, University of Seville, Camino de los Descubrimientos s/n, Seville, E-41092, Spain e-mail: [email protected] 1 Table 1 A Comparison of the FEM and BEM Finite element method Boundary element method Unknown DOFs everywhere Unknown DOFs only on boundary in domain for linear problems Banded symmetric matrices Smaller full nonsymmetric matrices Very efficient Less efficient in standard form (fast with new forms) Exterior problems somewhat Exterior problems in infinite domains difficult easy Nonhomogeneous problems Needs Green’s functions for maxi fine mum efficiency Moving boundary problems Efficient for moving boundary can be difficult problems Fig. 1 A 3D domain V with boundary S Needs compatible elements Incompatible=nonconforming ele ments fine Inter element discontinuity of Continuous internal derivatives derivatives 1.2 A Review of the BEM Formulation. We use the 3D potential theory problem as an example to show the key steps and main results in the BEM formulation. For a domain V with bound 1 Introduction ary S (Fig. 1), we have the following boundary value problem: The boundary element method (BEM) is a numerical method for r2/ðxÞ¼0; x 2 V (1) solving boundary value or initial value problems formulated in boundary integral equations (BIEs). In some literature, it is also where / is the potential field. The boundary conditions (BCs) are called the boundary integral equation method because of this rela tionship. The most important advantage of the BEM as compared /ðxÞ¼/ðxÞ; x 2 S/ ðDirichlet BCÞ (2) with other domain based numerical methods is the reduction of the @ dimensions of the problems to be solved, from 3D to 2D, or from 2D / qðxÞ ðxÞ¼qðxÞ; x 2 Sq ðNeumann BCÞ (3) to 1D, leading to much easier mesh generation as compared with @n domain based methods. The second advantage is the accuracy the BEM offers, due to the nature of integrals used in the formulations. in which the overbar indicates the prescribed value, S/ [ Sq ¼ S, The third advantage of the BEM is its capability of accurately model and n is the outward normal of the boundary S (Fig. 1). ing infinite domain problems without introducing additional, often The fundamental solution (also known as full space Green’s cumbersome, conditions at infinity. Table 1 is a comparison of the function) Gðx; yÞ for 3D potential problems is given by BEM with the finite element method (FEM), which may change in 1 the future based on new developments in either one or both methods. Gðx; yÞ¼ (4) 4pr 1.1 A Brief History. The direct BIE formulations (in which where r is the distance between source point x and field point y. the unknown functions have clear physical meanings) and their The normal derivative of G is modern numerical solutions with boundary elements originated more than 40 years ago during the 1960s. The 2D potential prob @Gðx; yÞ 1 Fðx; yÞ ¼ r; n ðyÞ (5) lem was first formulated in terms of a direct BIE and solved @nðyÞ 4pr2 k k numerically by Jaswon, et al. [1 3]. This work was later extended to the 2D elastostatic case by Rizzo in the early 1960s as his dis with r;k ¼ @r=@yk ¼ðyk xkÞ=r. The fundamental solution sertation work at the University of Illinois, which was later pub Gðx; yÞ represents the response (potential) at y due to a unit source lished as a journal article in 1967 [4]. Following these early at x (See Sec. 2.1 for the definition of a Green’s function). works, extensive research efforts were carried out in the BIE for If we apply the second Green’s identity with / and Gðx; yÞ in mulations of many problems in applied mechanics and in the nu domain V with boundary S, we obtain the following representation merical solutions during the 1960s and 1970s [5 20]. The name integral: boundary element method appeared in the middle of the 1970s in ð an attempt to make an analogy with the FEM [21 23]. /ðxÞ¼ ½Gðx; yÞqðyÞ Fðx; yÞ/ðyÞ dSðyÞ; x 2 V (6) Some personal accounts of the historical developments of the S BIE and BEM in various places of the U.S., Europe, and China Equation (6) is an expression of the solution / inside the domain were given by Rizzo [24], Cruse [25], Watson [26], Shippy [27], V for Eq. (1). Once the boundary values of both / and q are Mukherjee [28], Telles [29], and Yao and Du [30]inaspecialissue known on the entire boundary S, Eq. (6) can be applied to calcu of the Electronic Journal of Boundary Elements in 2003 [31]. A late / everywhere in V, if needed. To solve the unknown bound comprehensive review of the heritage and early history of the BIE ary values of / and q on S, we let x tend to the boundary S to and BEM was given by Cheng and Cheng in 2005 [32]. This review obtain the following BIE: provides vivid descriptions of the long history and major contribu ð tions to the mathematical foundations of the integral methods by cðxÞ/ðxÞ¼ ½Gðx; yÞqðyÞ Fðx; yÞ/ðyÞ dSðyÞ; x 2 S (7) the pioneers from the 18th to the first half of the 20th centuries and S the early days of the BEM until the late of 1970s. Another compre = hensive review of the BEM research up to the early of 1980s was in which cðxÞ is a coefficient and cðxÞ¼1 2ifS is smooth around given by Tanaka in 1983 [33]. A special review was given by x. Equation (7) is called a conventional or singular BIE for solving Tanaka, Sladek and Sladek in 1994 [34] on the research efforts in potential problems.
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