Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T08:18:25.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Isogeometric Analysis of the Dual Boundary Element Method for the Laplace Problem With a Degenerate Boundary

Published online by Cambridge University Press:  23 September 2019

J. H. Kao ,
K. H. Chen ,
J. T. Chen Open the ORCID record for J. T. Chen [Opens in a new window]  and
S. R. Kuo
J. H. Kao
Affiliation:
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan
K. H. Chen
Affiliation:
Department of Civil Engineering, National Ilan University, Ilan, Taiwan
J. T. Chen*
Affiliation:
Department of Harbor and River Engineering, Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, Taiwan Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan
S. R. Kuo
Affiliation:
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan
*
*Corresponding author ([email protected])
Get access

Abstract

In this paper, we develop the isogeometric analysis of the dual boundary element method (IGA-DBEM) to solve the potential problem with a degenerate boundary. The non-uniform rational B-Spline (NURBS) based functions are employed to interpolate the geometry and physical function. To deal with the rank-deficiency problem due to the degenerate boundary, the hypersingular integral equation is introduced to promote the full rank for the influence matrix in the dual BEM. Finally, three numerical examples are given to verify the accuracy of our proposed method. Both circular and square domains subjected to the Dirichlet boundary condition are considered. The engineering problem containing a degenerate boundary is considered, e.g., a seepage flow problem with a sheet pile. Numerical results of the IGA-DBEM agree well with these of the exact solution and the original dual boundary element method.

Keywords

Dual BEMIsogeometric analysisNURBSDegenerate boundaryLaplace equation
Type
Research Article
Information
Journal of Mechanics , Volume 36 , Issue 1 , February 2020 , pp. 35 - 46
Copyright
Copyright © 2019 The Society of Theoretical and Applied Mechanics 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Hughes, T.J.R., Cottrell, J.A., and Bazilevs, Y. “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement,” Computer Methods in Applied Mechanics and Engineering, 194, pp. 4135–4195 (2005).CrossRefGoogle Scholar
Simpson, R.N., Scott, M.A. Taus, M., Thomas, D.C. and Lian, H. “Acoustic isogeometric boundary element analysis,” Computer Methods in Applied Mechanics and Engineering, 269, pp. 265–290 (2014).CrossRefGoogle Scholar
Peake, M.J., Trevelyan, J. and Coates, G. “Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems,” Computer Methods in Applied Mechanics and Engineering, 284, pp. 762–780 (2015).CrossRefGoogle Scholar
Cottrell, J.A., Reali, A., Bazilevs, Y. and Hughes, T.J.R. “Isogeometric analysis of structural vibrations,” Computer Methods in Applied Mechanics and Engineering, 195, pp. 5257–5296 (2006).CrossRefGoogle Scholar
Auricchio, F., Beirao da Veiga, L., Buffa, A., Lovadina, C., Reali, A. and Sangalli, G. “A fully “locking-free” isogeometric approach for plane linear elasticity problems: A stream function formulation,” Computer Methods in Applied Mechanics and Engineering, 197, pp. 160–172 (2007).CrossRefGoogle Scholar
Schillinger, D., Dedé, L., Scott, M.A., Evans, J.A., Borden, M.J., Rank, E. and Hughes, T.J.R. “An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods and T-spline CAD surfaces,” Computer Methods in Applied Mechanics and Engineering, 249-252, pp. 116–150 (2012).CrossRefGoogle Scholar
Vázquez, R. and Buffa, A. “Isogeometric analysis for electromagnetic problems,” IEEE Transactions on Magnetics, 46, pp. 3305–3308 (2010).CrossRefGoogle Scholar
Buffa, A., Sangalli, G. and Vázquez, R. “Isogeometric analysis in electromagnetics: B-splines approximation,” Computer Methods in Applied Mechanics and Engineering, 199, pp. 1143–1152 (2010).CrossRefGoogle Scholar
Bazilevs, Y., Calo, V.M., Hughes, T.J.R. and Zhang, Y. “Isogeometric fluid-structure interaction: theory, algorithms, and computations,” Computational Mechanics, 43, pp. 3–37 (2008).CrossRefGoogle Scholar
Evans, J.A. and Hughes, T.J.R. “Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations,” Mathematical Models and Methods in Applied Sciences, 23, pp. 1421–1478 (2013).CrossRefGoogle Scholar
Heltai, L., Arroyo, M. and DeSimone, A. “Nonsingular isogeometric boundary element method for Stokes flows in 3D,” Computer Methods in Applied Mechanics and Engineering, 268, pp. 514–539 (2014).CrossRefGoogle Scholar
Rizzo, F.J. “An integral equation approach to boundary value problems in classical elastostatics,” Quarterly of Applied Mathematics, 25, pp. 83–95 (1967).CrossRefGoogle Scholar
He, W.J., Ding, H.J. and Hu, H.C. “Non-equivalence of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems,” Computers & Structures, 60, pp. 1029–1035 (1996).CrossRefGoogle Scholar
Chen, J.T., Chen, K.H., Chen, I.L., Liu, L.W. “A new concept of modal participation factor for numerical instability in the dual BEM for exterior acoustics,” Mechanics Research Communications, 30, pp. 161–174 (2003).CrossRefGoogle Scholar
Zhou, S.J., Sun, S.X. and Cao, X.Y. “The boundary contour method based on the equivalent boundary integral equation for 2-D linear elasticity,” Communications in Numerical Methods in Engineering, 15, pp. 811–821 (1999).Google Scholar
Constanda, C. “On the Dirichlet problem for the two dimensional biharmonic equation,” Mathematical Methods in the Applied Sciences, 20, pp. 885–890, (1997).3.0.CO;2-D>CrossRefGoogle Scholar
He, W.J. “An equivalent boundary integral formulation for bending problems of thin plates,” Computers & Structures, 74, pp. 319–322 (2000).CrossRefGoogle Scholar
Mitra, A.K. and Das, S. “Nonuniqueness in the integral equations formulation of the biharmonic equation in multiply connected domains,” Computer Methods in Applied Mechanics and Engineering, 69, pp. 205–214 (1988).CrossRefGoogle Scholar
Hong, H.-K., Chen, J.T. “Derivations of Integral Equations of Elasticity” Journal of Engineering Mechanics, 114, pp. 1028–1044 (1988).CrossRefGoogle Scholar
Chen, Y.Z., Lin, X.Y. “Regularity condition and numerical examination for degenerate scale problem of BIE for exterior problem of plane elasticity,” Engineering Analysis with Boundary Elements, 32, pp. 811–823 (2008).CrossRefGoogle Scholar
Chen, J.T., Wu, C.S., Chen, K.H. and Lee, Y.T. “Degenerate scale for analysis of circular plate using the boundary integral equations and boundary element method,” Computational Mechanics, 38, pp. 33–49 (2006).CrossRefGoogle Scholar
Ursell, F. “Irregular frequencies and the motion of floating bodies,” Journal of Fluid Mechanics, 105, pp. 143–156 (1981).CrossRefGoogle Scholar
Chen, J.T., Kuo, S.R. “On fictitious frequencies using circulants for radiation problems of a cylinder,” Mechanics Research Communications, 27, pp. 49–58 (2000).CrossRefGoogle Scholar
De Mey, G. “Calculation of eigenvalues of the Helmholtz equation by an integral equation,” International Journal for Numerical Methods in Engineering, 10, pp. 59–66 (1976).CrossRefGoogle Scholar
Chen, J.T., Liu, L.W. and Hong, H.-K. “Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply-connected problem,” Proceedings of the Royal Society of London A, 459, pp. 1891–1925 (2003).CrossRefGoogle Scholar
Chen, J.T., Lin, S.R., Chen, K.H., Chen, I.L. and Chyuam, S.W. “Eigenanalysis for membranes with stringers using conventional BEM in conjunction with SVD technique,” Computer Methods in Applied Mechanics and Engineering, 192, pp. 1299–1322 (2003).CrossRefGoogle Scholar
Terai, T. “On calculation of sound fields around three dimensional objects by integral equation methods,” Journal of Sound and Vibration, 69, pp. 799–823 (1980).CrossRefGoogle Scholar
De Lacerda, L.A., Wrobel, L.C., Power, H. and Mansur, W.J. “A novel boundary integral formulation for three-dimensional analysis of thin acoustic barriers over an impedance plane,” Journal of the Acoustical Society of America, 104, pp. 671–678 (1998).CrossRefGoogle Scholar
Portela, A., Aliabadi, M.H. and Rooke, D.P. “The dual boundary element method: effective implementation for crack problems,” International Journal for Numerical Methods in Engineering, 33, pp. 1269–1287 (1992).CrossRefGoogle Scholar
Tuck, E.O., Application and Numerical Solution of Cauchy-Singular Integral Equations, The Application and Numerical Solution of Integral Equations, Edited by Anderssen, R. S., de Hoog, F. R. , and Lukas, M. A., Sijthoff and Noordhoff, The Netherlands, pp. 21–49 (1980).CrossRefGoogle Scholar
Mangler, K.W., Improper integrals in theoretical aerodynamics, Nine Shillings Net, London (1951).Google Scholar
Martin, P.A., Rizzo, F.J. and Gonsalves, I.R. “On hypersingular integral equations for certain problems in mechanics,” Mechanics Research Communications, 16, pp. 65–71 (1989).CrossRefGoogle Scholar
Chen, J.T. and Hong, H.-K. “Review of dual integral representations with emphasis on hypersingular integrals and divergent series,” Applied Mechanics Reviews, 52, pp. 17–33 (1999).CrossRefGoogle Scholar
Chen, J.T., Huang, W.S., Fan, Y., Kao, S.K. “Revisit of dual BEM using SVD updating technique,” Journal of Mechanics, 31, pp. 505–514 (2015).CrossRefGoogle Scholar
Chen, K.H., Chen, J.T., Chou, C.R. and Yueh, C.Y. “Dual boundary element analysis of oblique incident wave passing a thin submerged breakwater,” Engineering Analysis with Boundary elements, 26, pp.917–928 (2002).CrossRefGoogle Scholar
Chen, K.H., Chen, J.T., Lin, S.Y. and Lee, Y.T. “Dual boundary element analysis of normal incident wave passing a thin submerged breakwater with rigid, absorbing, and permeable boundaries,” Journal of Waterway, Port, Costal and Ocean Engineering, ASCE, 130, pp.179–190 (2004).CrossRefGoogle Scholar
Hsieh, M. C. “The study of wave damping characteristics and wave forces on submerged plates,” Master Thesis, Department of Harbor and River Engineering, National Taiwan Ocean University, Taiwan (2004). (in Chinese)Google Scholar
Wu, W.P., “Experimental study of wave propagating a series of submerged horizontal plates,” Master Thesis, Department of Harbor and River Engineering, National Taiwan Ocean University, Taiwan (2004). (in Chinese)Google Scholar
Lafe, O.E., Montes, J.S., Cheng, A.H.D., Liggett, J.A. and Liu, L.-F. “Singularities in Darcy flow through porous media,” Journal of the Hydraulics Division, 106, pp. 977–997 (1980).Google Scholar
Chen, J.T., Lin, S.R., Chen, K.H., Chen, I.L., and Chyuan, S.W. “Eigenanalysis for membranes with stringers using conventional BEM in conjunction with SVD technique,” Computer Methods in Applied Mechanics and Engineering, 192, pp. 1299–1322 (2003).CrossRefGoogle Scholar
Sun, F.L., Dong, C.Y., Yang, H.S., “Isogeometric boundary element method for crack propagation based on Bézier extraction of NURBS,” Engineeing Analysis with Boundary Elements, 99, pp. 76–88 (2019).CrossRefGoogle Scholar
Peddie, J., The History of Visual Magic in Computers: How beautiful images are made in CAD, 3D, VR and AR, Springer, New York (2013).Google Scholar
De Boor, C., A Practical Guide to Splines, Revised Edition, Springer, New York (1978).CrossRefGoogle Scholar
Simpson, R.N., Bordas, S.P.A., Trevelyan, J., and Rabczuk, T.A. “Two dimensional Isogeometric Boundary Element Method for elastostatic analysis,” Computer Methods in Applied Mechanics and Engineering, 209, pp. 87–100 (2012).CrossRefGoogle Scholar
Cottrell, J.A., Hughes, T.J.R., and Bazilevs, Y., Isogeometric Analysis. Toward Integration of CAD and FEA, Wiley, UK (2009).CrossRefGoogle Scholar
Chen, J.T., Chen, Y.W., “Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary,” Engineeing Analysis with Boundary Elements, 24, pp. 671–684 (2000).CrossRefGoogle Scholar