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An element implementation of the boundary face method for 3D potential problems. (English) Zbl 1244.74182

Summary: This work presents a new implementation of the boundary face method (BFM) with shape functions from surface elements on the geometry directly like the boundary element method (BEM). The conventional BEM uses the standard elements for boundary integration and approximation of the geometry, and thus introduces errors in geometry. In this paper, the BFM is implemented directly based on the boundary representation data structure (B-rep) that is used in most CAD packages for geometry modeling. Each bounding surface of geometry model is represented as parametric form by the geometric map between the parametric space and the physical space. Both boundary integration and variable approximation are performed in the parametric space. The integrand quantities are calculated directly from the faces rather than from elements, and thus no geometric error will be introduced. The approximation scheme in the parametric space based on the surface element is discussed. In order to deal with thin and slender structures, an adaptive integration scheme has been developed. An adaptive method for generating surface elements has also been developed. We have developed an interface between BFM and UG-\(NX(R)\). Numerical examples involving complicated geometries have demonstrated that the integration of BFM and UG-\(NX(R)\) is successful. Some examples have also revealed that the BFM possesses higher accuracy and is less sensitive to the coarseness of the mesh than the BEM.

MSC:

74S15 Boundary element methods applied to problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs

Software:

NX
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Zhang, J. M.; Qin, X. Y.; Han, X.; Li, G. Y., A boundary face method for potential problems in three dimensions, Int J Numer Meth Eng, 80, 320-337 (2009) · Zbl 1176.74212
[2] Mukherjee, Y. X.; Mukherjee, S., The boundary node method for potential problems, Int J Numer Meth Eng, 40, 797-815 (1997) · Zbl 0885.65124
[3] Zhang, J. M.; Yao, Z. H.; Li, H., A hybrid boundary node method, Int J Numer Meth Eng, 53, 751-763 (2002)
[4] Liu, G. R.; Gu, Y. T., An introduction to mesh free methods and their programming (2005), Springer: Springer Dordrecht, The Netherlands, USA
[5] Liu, C. S., An effectively modified direct Trefftz method for 2D potential problems considering the domain’s characteristic length, Eng Anal Bound Elem, 31, 983-993 (2007) · Zbl 1259.65183
[6] Kagan, P.; Fischer, A., Integrated mechanically based CAE system using B-Spline finite elements, Comput-Aid Design, 32, 539-552 (2000) · Zbl 1206.65050
[7] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput Methods Appl Mech Eng, 194, 4135-4195 (2005) · Zbl 1151.74419
[8] Wang, L., Integration of CAD and boundary element analysis through subdivision methods, \(C\) omput Ind Eng, 57, 3, 691-698 (2009)
[9] Lee, C. K.; Hobbs, R. E., Automatic adaptive finite element mesh generation over rational B-spline surfaces, Comput Struct, 69, 577-608 (1998) · Zbl 0941.74067
[10] Miranda, A. C.; Meggiolaro, M. A., Castro JTP. Fatigue life and crack path predictions in generic 2D structural components, Eng Fract Mech, 70, 1259-1279 (2003)
[11] Shaw, Amit; Roy, D., NURBS-based parametric mesh-free methods, Comput Methods Appl Mech Eng, 197, 1541-1567 (2008) · Zbl 1194.74538
[12] Kane, J. H., Boundary element analysis in engineering continuum mechanics (1994), Prentice-Hall Inc: Prentice-Hall Inc Englewood Cliffs
[13] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J. A.; Hughes, T. J.R., Isogeometric analysis using T-splines, Comput Methods Appl Mech Eng, 199, 229-263 (2010) · Zbl 1227.74123
[14] Chati, M. K.; Mukherjee, S., The boundary node method for three-dimensional problems in potential theory, Int J Numer Meth Eng, 47, 1523-1547 (2000) · Zbl 0961.65100
[15] Nagaranjan, A.; Mukherjee, S., A mapping method for numerical evaluation of two-dimensional integrals with \(1/r\) singularity, Comput Mech, 12, 19-26 (1993) · Zbl 0776.73073
[16] Liu, Y. J., Analysis of shell-like structures by the boundary element method based on 3-D elasticity: formulation and verification, Int J Numer Meth Eng, 41, 541-558 (1998) · Zbl 0910.73068
[17] Luo, J. F.; Liu, Y. J.; Berger, E. J., Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method, Comput Mech, 22, 404-412 (1998) · Zbl 0938.74075
[18] Chen, X. L.; Liu, Y. J., An advanced 3-D boundary element method for characterizations of composite materials, Eng Anal Bound Elem, 29, 6, 513-523 (2005) · Zbl 1182.74212
[19] Liu, Y. J.; Zhang, D. M.; Rizzo, F. J., Nearly singular and hypersingular integrals in the boundary element method, (Brebbia, C. A.; Rencis, J. J., Boundary elements XV (1993), Computational Mechanics Publications: Computational Mechanics Publications Worcester, MA), 453-468
[20] Niu, Z. R.; Cheng, C. Z.; Zhou, H. L.; Hu, Z. J., Analytic formulations for calculating nearly singular integrals in two-dimensional BEM, Eng Anal Bound Elem, 31, 949-964 (2007) · Zbl 1259.74056
[21] Chen, H. B.; Lu, P.; Huang, M. G.; Williams, F. W., An effective method for finding values on and near boundaries in the elastic BEM, Comput Struct, 69, 4, 421-431 (1998) · Zbl 0941.74075
[22] Chen, J. T.; Hong, H. K., Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series, Appl Mech Rev, 52, 1, 17-33 (1999)
[23] Gao, X. W.; Yang, K.; Wang, J., An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals, Eng Anal Bound Elem, 32, 692-696 (2008) · Zbl 1244.65199
[24] Zhang, J. M.; Masa, Tanaka; Endo, M., The hybrid boundary node method accelerated by fast multipole method for 3D potential problems, Int J Numer Meth Eng, 63, 660-680 (2005) · Zbl 1085.65115
[25] Zhang, J. M.; Masa, Tanaka, Adaptive spatial decomposition in fast multipole method, J Comput Phys, 226, 17-28 (2007) · Zbl 1124.65115
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