An isogeometric BEM for exterior potential-flow problems in the plane

In this paper, the isogeometric concept introduced by Hughes, in the context of Finite Element Method, is applied to Boundary Element Method (BEM), for solving an exterior planar Neumann problem. The developed isogeometric-BEM concept is based on NURBS, for representing the exact body geometry and employs the same basis for representing the potential and/or the density of the single layer. In order to examine the accuracy of the scheme, numerical results for the case of a circle and a free-form body are presented and compared against analytical solutions. This enables performing a numerical error analysis, verifying the superior convergence rate of the isogeometric BEM versus low-order BEM. When starting from the initial NURBS representation of the geometry and then using knot insertion for refinement of the NURBS basis, the achieved rate of convergence is O(DoF-4). This rate may be further improved by using a degree-elevated initial NURBS representation of the geometry (kh-refinement).

[1]  L. Eça,et al.  A numerical study on low and higher-order potential based BEM for 2D inviscid flows , 2003 .

[2]  S. Kinnas,et al.  Re-Entrant Jet Modelling for Partially Cavitating Two-Dimensional Hydrofoils , 2001 .

[3]  Thomas J. R. Hughes Isogeometric Analysis : Progress and Challenges , 2008 .

[4]  J. Sládek,et al.  Numerical integration of logarithmic and nearly logarithmic singularity in BEMs , 2001 .

[5]  C. Schwab,et al.  Boundary Element Methods , 2010 .

[6]  F. Hartmann,et al.  BOUNDARY ELEMENT METHODS , 2001 .

[7]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[8]  C. Pozrikidis,et al.  Interfacial dynamics for Stokes flow , 2001 .

[9]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

[10]  G. Schmidt,et al.  The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian , 1988 .

[11]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[12]  M. A. Jaswon,et al.  Integral equation methods in potential theory and elastostatics , 1977 .

[13]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[14]  M. Kropinski An efficient numerical method for studying interfacial motion in two-dimensional creeping flows , 2001 .

[15]  D. M. Friedman Improved solution for potential flow about arbitrary axisymmetric bodies by the use of a higher-order surface source method. Part 2. User's manual for computer program , 1974 .

[16]  Carlos Alberto Brebbia Recent innovations in BEM , 2002 .

[17]  T. Hou,et al.  Removing the stiffness from interfacial flows with surface tension , 1994 .

[18]  G. Farin Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[19]  Les A. Piegl,et al.  The NURBS book (2nd ed.) , 1997 .

[20]  L. M. Milne-Thomson,et al.  Theoretical hydrodynamics / by L.M. Milne-Thomson , 1955 .

[21]  R. Kress Linear Integral Equations , 1989 .