zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A two-dimensional isogeometric boundary element method for elastostatic analysis. (English) Zbl 1243.74193
Summary: The concept of isogeometric analysis, where functions that are used to describe geometry in CAD software are used to approximate the unknown fields in numerical simulations, has received great attention in recent years. The method has the potential to have profound impact on engineering design, since the task of meshing, which in some cases can add significant overhead, has been circumvented. Much of the research effort has been focused on finite element implementations of the isogeometric concept, but at present, little has been seen on the application to the Boundary Element Method. The current paper proposes an Isogeometric Boundary Element Method (BEM), which we term IGABEM, applied to two-dimensional elastostatic problems using Non-Uniform Rational B-Splines (NURBS). We find it is a natural fit with the isogeometric concept since both the NURBS approximation and BEM deal with quantities entirely on the boundary. The method is verified against analytical solutions where it is seen that superior accuracies are achieved over a conventional quadratic isoparametric BEM implementation.

74S15Boundary element methods in solid mechanics
65N38Boundary element methods (BVP of PDE)
65D17Computer aided design (modeling of curves and surfaces)
65T60Wavelets (numerical methods)
74F15Electromagnetic effects in solid mechanics
Full Text: DOI
[1] <https://github.com/bobbiesimpson/isogeometric-bem>, 2011.
[2] Auricchio, F.; Da Veiga, L. B.; Hughes, T. J. R.; Reali, A.; Sangalli, G.: Isogeometric collocation methods, Math. models methods appl. Sci. 20, No. 11, 2075-2107 (2010) · Zbl 1226.65091 · doi:10.1142/S0218202510004878
[3] Barber, J. R.: Elasticity, (2002)
[4] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J. A.; Hughes, T. J. R.; Lipton, S.; Scott, M. A.; Sederberg, T. W.: Isogeometric analysis using T-splines, Comput. methods appl. Mech. engrg. 199, No. 5 -- 8, 229-263 (2010) · Zbl 1227.74123 · doi:10.1016/j.cma.2009.02.036
[5] Cruse, T.: Numerical solutions in three dimensional elastostatics, Int. J. Solids struct. 5, 1259-1274 (1969) · Zbl 0181.52404 · doi:10.1016/0020-7683(69)90071-7
[6] Deng, J.; Chen, F.; Li, X.; Hu, C.; Tong, W.; Yang, Z.; Feng, Y.: Polynomial splines over hierarchical T-meshes, Graph. models 70, 76-86 (2008)
[7] Greville, T.: Numerical procedures for interpolation by spline functions, J. soc. Ind. appl. Math. ser. B. numer. Anal. (1964) · Zbl 0141.33602
[8] Guiggiani, M.; Casalini, P.: Direct computation of Cauchy principal value integrals in advanced boundary elements, Int. J. Numer. methods engrg. 24, 1711-1720 (1987) · Zbl 0635.65020 · doi:10.1002/nme.1620240908
[9] Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. methods appl. Mech. engrg. 194, No. 39 -- 41, 4135-4195 (2005) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[10] Hughes, T. J. R.; Reali, A.; Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis, Comput. methods appl. Mech. engrg. 199, No. 5 -- 8, 301-313 (2010) · Zbl 1227.65029 · doi:10.1016/j.cma.2008.12.004
[11] Johnson, R.: Higher order B-spline collocation at the greville abscissae, Appl. numer. Math. 52, 63-75 (2005) · Zbl 1063.65072 · doi:10.1016/j.apnum.2004.04.002
[12] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M.: Meshless methods: a review and computer implementation aspects, Math. comput. Simul. 79, No. 3, 763-813 (2008) · Zbl 1152.74055 · doi:10.1016/j.matcom.2008.01.003
[13] Nguyen-Thanh, N.; Nguyen-Xuan, H.; Bordas, S. P. A.; Rabczuk, T.: Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids, Comput. methods appl. Mech. engrg., 1-37 (2011) · Zbl 1228.74091
[14] Piegl, L.; Tiller, W.: The NURBS book, (1995) · Zbl 0828.68118
[15] Rizzo, F. J.: An integral equation approach to boundary value problems of classical elastostatics, Q. appl. Math. 25, No. 1, 83-95 (1967) · Zbl 0158.43406
[16] Rizzo, F. J.; Shippy, D. J.: An advanced boundary integral equation method for three-dimensional thermoelasticity, Int. J. Numer. methods engrg. 11, 1753-1768 (1977) · Zbl 0387.73007 · doi:10.1002/nme.1620111109
[17] Rogers, D. F.: An introduction to NURBS: with historical perspective, (2001)
[18] Simpson, R.; Trevelyan, J.: A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics, Comput. methods appl. Mech. engrg. 200, No. 1 -- 4, 1-10 (2010) · Zbl 1225.74117 · doi:10.1016/j.cma.2010.06.015
[19] Szabó, B.; Babuška, I.: Finite element analysis, (1991) · Zbl 0792.73003
[20] Telles, J. C. F.: A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary integrals, Int. J. Numer. methods engrg. 24, 959-973 (1987) · Zbl 0622.65014 · doi:10.1002/nme.1620240509
[21] Trevelyan, J.; Coates, G.: On adaptive definition of the plane wave basis for wave boundary elements in acoustic scattering: the 2D case, Comput. model. Engrg. sci. 55, 147-170 (2010) · Zbl 1231.76287 · doi:10.3970/cmes.2010.055.147
[22] Uhm, T. -K.; Kim, K. -S.; Seo, Y. -D.; Youn, S. -K.: A locally refinable T-spline finite element method for CAD/CAE integration, Struct. engrg. Mech. 30, No. 2, 225-245 (2008)
[23] Uhm, T. -K.; Youn, S. -K.: T-spline finite element method for the analysis of shell structures, Int. J. Numer. methods engrg. 80, No. 4, 507-536 (2009) · Zbl 1176.74198 · doi:10.1002/nme.2648
[24] Watson, J. O.: Advanced implementation of the boundary element method for two- and three-dimensional elastostatics, (1979) · Zbl 0451.73075