×

Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems. (English) Zbl 1439.74451

Summary: The isogeometric analysis (IGA) is applied for the weakly singular symmetric Galerkin boundary element method (SGBEM) to analyzelinear elastostatics problems in three-dimensional domains. The background of the proposed method is to use non-uniform rational B-splines (NURBS) as the basis functions for the approximation of both geometry and field variables (i.e. displacement and traction) of the governing integral equations. Same as weakly singular SGBEM, the basic ingredient of the method is a pair of weakly singular weak-form integral equations for the displacement and traction on the boundary of the domain. These integral equations are solved approximately using standard Galerkin approximation. In addition to the advantages that IGA owned, the proposed method exploits the common boundary representation of CAD model and boundary element method. Various numerical examples of both simple and complex geometries are examined to validate the accuracy and efficiency of the proposed method. Through the numerical examples, it is observed that the IGA-SGBEM produces highly accurate results.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
74B05 Classical linear elasticity

Software:

BEAN; GeoPDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cottrell, J.; Hughes, T.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), John Wiley & Sons Inc. · Zbl 1378.65009
[2] Hughes, T.; Cottrell, J.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 39-41, 4135-4195 (2005) · Zbl 1151.74419
[3] Nguyen-Thanh, N.; Kiendl, J.; Nguyen-Xuan, H.; Wüchner, R.; Bletzinger, K.; Bazilevs, Y.; Rabczuk, T., Rotation free isogeometric thin shell analysis using PHT-splines, Comput. Methods Appl. Mech. Engrg., 200, 47-48, 3410-3424 (2011) · Zbl 1230.74230
[4] Nguyen-Thanh, N.; Valizadeh, N.; Nguyen, M.; Nguyen-Xuan, H.; Zhuang, X.; Areias, P.; Zi, G.; Bazilevs, Y.; Lorenzis, L. D.; Rabczuk, T., An extended isogeometric thin shell analysis based on kirchhoff-love theory, Comput. Methods Appl. Mech. Engrg., 284, 265-291 (2015) · Zbl 1423.74811
[5] Brebbia, C. A.; Dominguez, J., Boundary Elements: an Introductory Course (1992), WIT Press · Zbl 0780.73002
[6] Gu, J.; Zhang, J.; Li, G., Isogeometric analysis in BIE for 3-D potential problem, Eng. Anal. Bound. Elem., 36, 5, 858-865 (2012) · Zbl 1352.65585
[7] Li, K.; Qian, X., Isogeometric analysis and shape optimization via boundary integral, Comput. Aided Des., 43, 11, 1427-1437 (2011)
[8] Simpson, R.; Bordas, S.; Trevelyan, J.; Rabczuk, T., A two-dimensional isogeometric boundary element method for elastostatic analysis, Comput. Methods Appl. Mech. Engrg., 209-212, 87-100 (2012) · Zbl 1243.74193
[9] Bazilevs, Y.; Calo, V.; Cottrell, J.; Evans, J.; Hughes, T.; Lipton, S.; Scott, M.; Sederberg, T., Isogeometric analysis using t-splines, Comput. Methods Appl. Mech. Engrg., 199, 5-8, 229-263 (2010) · Zbl 1227.74123
[10] Scott, M.; Simpson, R.; Evans, J.; Lipton, S.; Bordas, S.; Hughes, T.; Sederberg, T., Isogeometric boundary element analysis using unstructured t-splines, Comput. Methods Appl. Mech. Engrg., 254, 197-221 (2013) · Zbl 1297.74156
[11] Simpson, R.; Scott, M.; Taus, M.; Thomas, D.; Lian, H., Acoustic isogeometric boundary element analysis, Comput. Methods Appl. Mech. Engrg., 269, 265-290 (2014) · Zbl 1296.65175
[12] Ginnis, A.; Kostas, K.; Politis, C.; Kaklis, P.; Belibassakis, K.; Gerostathis, T.; Scott, M.; Hughes, T., Isogeometric boundary-element analysis for the wave-resistance problem using t-splines, Comput. Methods Appl. Mech. Engrg., 279, 425-439 (2014) · Zbl 1423.74270
[13] Wang, Y.; Benson, D. J.; Nagy, A. P., A multi-patch nonsingular isogeometric boundary element method using trimmed elements, Comput. Mech., 56, 1, 173-191 (2015) · Zbl 1329.65281
[14] Beer, G.; Marussig, B.; Zechner, J., A simple approach to the numerical simulation with trimmed CAD surfaces, Comput. Methods Appl. Mech. Engrg., 285, 776-790 (2015) · Zbl 1425.65029
[15] Beer, G., Advanced Numerical Simulation Methods: From CAD Data Directly to Simulation Results (2015), CRC Press · Zbl 1344.65021
[16] Peng, X.; Atroshchenko, E.; Kerfriden, P.; Bordas, S., Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth, Comput. Methods Appl. Mech. Engrg. (2016) · Zbl 1439.74370
[17] Bui, H., An integral equations method for solving the problem of a plane crack of arbitrary shape, J. Mech. Phys. Solids, 25, 1, 29-39 (1977) · Zbl 0355.73074
[18] Bonnet, M.; Maier, G.; Polizzotto, C., Symmetric galerkin boundary element methods, Appl. Mech. Rev., 51, 11, 669-704 (1998)
[19] Sutradhar, A.; Paulino, G.; Gray, L., Symmetric Galerkin Boundary Element Method (2008), Springer, Berlin, Heidelberg · Zbl 1156.65101
[20] Li, S.; Mear, M. E., Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media, Int. J. Fract., 93, 1/4, 87-114 (1998)
[21] Li, S.; Mear, M.; Xiao, L., Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Methods Appl. Mech. Engrg., 151, 3, 435-459 (1998) · Zbl 0906.73074
[22] Rungamornrat, J.; Mear, M. E., A weakly-singular SGBEM for analysis of cracks in 3D anisotropic media, Comput. Methods Appl. Mech. Engrg., 197, 49-50, 4319-4332 (2008) · Zbl 1194.74501
[23] Rungamornrat, J.; Mear, M. E., Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media, Int. J. Solids Struct., 45, 5, 1283-1301 (2008) · Zbl 1169.74549
[24] Tran, H. D.; Mear, M. E., Regularized boundary integral equations for two-dimensional crack problems in multi-field media, Int. J. Fract., 181, 1, 99-113 (2013)
[25] Tran, H. D.; Mear, M. E., A weakly singular SGBEM for analysis of two-dimensional crack problems in multi-field media, Eng. Anal. Bound. Elem., 45, 60-73 (2014) · Zbl 1297.74158
[26] Aimi, A.; Diligenti, M.; Sampoli, M.; Sestini, A., Isogemetric analysis and symmetric galerkin BEM: A 2D numerical study, Appl. Math. Comput. (2015) · Zbl 1410.65468
[27] Aimi, A.; Diligenti, M.; Sampoli, M.; Sestini, A., Non-polynomial spline alternatives in isogeometric symmetric galerkin BEM, Appl. Numer. Math. (2016) · Zbl 1372.65316
[28] Nguyen, B.; Tran, H.; Anitescu, C.; Zhuang, X.; Rabczuk, T., An isogeometric symmetric galerkin boundary element method for two-dimensional crack problems, Comput. Methods Appl. Mech. Engrg., 306, 252-275 (2016) · Zbl 1436.74083
[29] Xiao, L., Symmetric Weak-form Integral Equation Method for Three Dimensional Fracture Analysis (1998), University of Texas at Austin · Zbl 0906.73074
[30] Les Piegl, W. T., The NURBS Book (1997), Springer
[31] Nguyen, V. P.; Anitescu, C.; Bordas, S. P.; Rabczuk, T., Isogeometric analysis: An overview and computer implementation aspects, Math. Comput. Simulation, 117, 89-116 (2015) · Zbl 07313396
[32] Vázquez, R., A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0, Comput. Math. Appl., 72, 3, 523-554 (2016) · Zbl 1359.65270
[33] Timoshenko, S.; Goodier, J., Theory of Elasticity (1969), McGraw-Hill · Zbl 0045.26402
[34] Nguyen, V. P.; Kerfriden, P.; Brino, M.; Bordas, S. P.A.; Bonisoli, E., Nitsche’s method for two and three dimensional NURBS patch coupling, Comput. Mech., 53, 6, 1163-1182 (2013) · Zbl 1398.74379
[35] Brino, M., CAD-CAE Integration and Isogeometric Analysis: Trivariate Multipatch and Applications (2015), Politecnico di Torino, (Ph.D. thesis)
[36] SIMO-Package, 2016. https://github.com/SIMOGroup/SIMO-Package; SIMO-Package, 2016. https://github.com/SIMOGroup/SIMO-Package
[37] Kiendl, J.; Bazilevs, Y.; Hsu, M.-C.; Wüchner, R.; Bletzinger, K.-U., The bending strip method for isogeometric analysis of kirchhoff-love shell structures comprised of multiple patches, Comput. Methods Appl. Mech. Engrg., 199, 37-40, 2403-2416 (2010) · Zbl 1231.74482
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.