zbMATH — the first resource for mathematics

An isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution. (English) Zbl 1440.74286
Summary: A study of structural shape optimization and absorbing material distribution optimization of noise barrier structures based on the recently proposed isogeometric analysis method with exact geometric definitions is presented. The acoustic scattering is approximated using the basis functions that represent the geometry. A fast multipole method is applied to accelerate the solution of the boundary element method (BEM). The Burton-Miller formulation is used to overcome the fictitious frequency problem when a single Helmholtz boundary integral equation is used for the exterior boundary-value problem. The strongly singular integrals in the Burton-Miller formulation using the isogeometric BEM are evaluated explicitly and directly, particularly for the sensitivity formulation with a hyper-singular integral. The optimality criteria method is used for two types of optimization analyses, shape optimization and material distribution topology optimization. For the shape optimization, the design variables can be set to the locations of the control points because the control points determine the shape of structure. For the second optimization, a new material interpolation scheme for acoustic problems based on the solid isotropic material with penalization (SIMP) method is given, where the interpolation variable is not the real structural density used in a conventional SIMP, but a fictitious material density that determines the normalized surface admittance. Several examples are presented to demonstrate the validity and efficiency of the proposed algorithm.

74P15 Topological methods for optimization problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
74S15 Boundary element methods applied to problems in solid mechanics
PDF BibTeX Cite
Full Text: DOI
[1] 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, 39, 4135-4195 (2005) · Zbl 1151.74419
[2] Takahashi, T.; Matsumoto, T., An application of fast multipole method to isogeometric boundary element method for Laplace equation in two dimensions, Eng. Anal. Bound. Elem., 36, 1766-1775 (2012) · Zbl 1351.74138
[3] Gu, J.; Zhang, J.; Li, G., Isogeometric analysis in BIE for 3-D potential problem, Eng. Anal. Bound. Elem., 36, 858-865 (2012) · Zbl 1352.65585
[4] Gong, Y.; Dong, C.; Qin, X., An isogeometric boundary element method for three dimensional potential problems, J. Comput. Appl. Math., 313, 454-468 (2017) · Zbl 1353.65129
[5] Auricchio, F.; Beirao da Veiga, L.; Buffa, A.; Lovadina, C.; Reali, A.; Sangalli, G., A fully “locking-free” isogeometric approach for plane linear elasticity problems: a stream function formulation, Comput. Methods Appl. Mech. Engrg., 197, 1, 160-172 (2007) · Zbl 1169.74643
[6] Schillinger, D.; Dedé, L.; Scott, M. A.; Evans, J. A.; Borden, M. J.; Rank, E.; 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, Comput. Methods Appl. Mech. Eng., 249-252, 116-150 (2012) · Zbl 1348.65055
[7] Simpson, R. N.; Bordas, S. P.A.; 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
[8] Yoon, M.; Cho, S., Isogeometric shape design sensitivity analysis of elasticity problems using boundary integral equations, Eng. Anal. Bound. Elem., 66, 119-128 (2016) · Zbl 1403.74262
[9] Bai, Y.; Dong, C.; Liu, Z., Effective elastic properties and stress states of doubly periodic array of inclusions with complex shapes by isogeometric boundary element method, Compos. Struct., 128, 54-69 (2015)
[10] Peng, X.; Atroshchenko, E.; Kerfriden, P.; Bordas, S., Linear elastic fracture simulation directly from CAD: 2D NURBS-based implementation and role of tip enrichment, Int. J. Fract., 204, 55-78 (2017)
[12] Vázquez, R.; Buffa, A., Isogeometric analysis for electromagnetic problems, IEEE Trans. Magn., 46, 8, 3305-3308 (2010)
[13] Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Zhang, Y., Isogeometric fluid-structure interaction: theory, algorithms, and computations, Comput. Mech., 43, 1, 3-37 (2008) · Zbl 1169.74015
[14] Evans, J. A.; Hughes, T. J.R., Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations, Math. Models Methods Appl. Sci., 23, 08, 1421-1478 (2013) · Zbl 1383.76337
[15] Heltai, L.; Arroyo, M.; DeSimone, A., Nonsingular isogeometric boundary element method for Stokes flows in 3D, Comput. Methods Appl. Mech. Engrg., 268, 514-539 (2014) · Zbl 1295.76022
[16] Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Engrg., 195, 41, 5257-5296 (2006) · Zbl 1119.74024
[17] Peake, M. J.; Trevelyan, J.; Coates, G., Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems, Comput. Methods Appl. Mech. Eng., 259, 93-102 (2013) · Zbl 1286.65176
[18] Simpson, R. N.; Scott, M. A.; Taus, M.; Thomas, D. C.; Lian, H., Acoustic isogeometric boundary element analysis, Comput. Methods Appl. Mech. Eng., 269, 265-290 (2014) · Zbl 1296.65175
[19] Simpson, R.; Liu, Z., Acceleration of isogeometric boundary element analysis through a black-box fast multipole method, Eng. Anal. Bound. Elem., 66, 168-182 (2016) · Zbl 1403.65228
[20] Peake, M.; Trevelyan, J.; Coates, G., Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems, Comput. Methods Appl. Mech. Engrg., 284, 762-780 (2015) · Zbl 1425.65202
[21] Marburg, S., Developments in structural-acoustic optimisation for passive noise control, Arch. Comput. Methods Eng., 27, 291-370 (2002) · Zbl 1099.74538
[22] Chen, L.; Zheng, C.; Chen, H., A wideband FMBEM for 2D acoustic design sensitivity analysis based on direct differentiation method, Comput. Mech., 52, 6, 631-648 (2013) · Zbl 1282.74101
[23] Chen, L.; Zheng, C.; Chen, H., FEM/wideband FMBEM coupling for structural-acoustic design sensitivity analysis, Comput. Methods Appl. Mech. Engrg., 276, 12, 1-19 (2014) · Zbl 1423.74263
[24] Chen, L.; Chen, H.; Zheng, C.; Marburg, S., Structural-acoustic sensitivity analysis of radiated sound power using a finite element/discontinuous fast multipole boundary element scheme, Internat. J. Numer. Methods Fluids, 82, 12, 858-878 (2016)
[25] Chen, L.; Liu, L.; Zhao, W.; Chen, H., 2D acoustic design sensitivity analysis based on adjoint variable method using different types of boundary elements, Acoust. Aust., 44, 12, 343-357 (2016)
[26] Chen, L.; Marburg, S.; Chen, H.; Zhang, H.; Gao, H., An adjoint operator approach for sensitivity analysis of radiated sound power in fully coupled structural-acoustic systems, J. Comput. Acoust., 25, 1750003 (2017)
[27] Zheng, C.; Zhang, C.; Bi, C.; Gao, H.; Du, L.; Chen, H., Coupled FE-BE method for eigenvalue analysis of elastic 359 structures submerged in an infinite fluid domain, Internat. J. Numer. Methods Engrg., 110, 163-185 (2017) · Zbl 1365.74072
[28] Li, K.; Qian, X., Isogeometric analysis and shape optimisation via boundary integral, Comput. Aided Des., 43, 11, 1427-1437 (2011)
[29] Lian, H.; Kerfriden, P.; Bordas, S., Implementation of regularized isogeometric boundary element methods for gradient-based shape optimisation in two-dimensional linear elasticity, Internat. J. Numer. Methods Engrg., 106, 12, 972-1017 (2016) · Zbl 1352.74467
[30] Lian, H.; Kerfriden, P.; Bordas, S., Shape optimization directly from CAD: An isogeometric boundary element approach using T-splines, Comput. Methods Appl. Mech. Engrg., 317, 1-41 (2017)
[31] Kostasa, K.; Ginnis, A.; Politis, C.; Kaklis, P., Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver, Comput. Aided Des., 82, 79-87 (2017)
[32] Lee, S.; Yoon, M.; Cho, S., Isogeometric topological shape optimization using dual evolution with boundary integral equation and level sets, Comput. Aided Des., 82, 88-99 (2017)
[33] Bendsøe, M., Optimal shape design as a material distribution problem, Struct. Optim., 1, 4, 193-202 (1989)
[34] Toledo, R.; Aznárez, J.; Maeso, O.; Greiner, D., Optimization of thin noise barrier designs using evolutionary algorithms and a dual BEM formulation, J. Sound Vib., 334, 219-238 (2015)
[35] Toledo, R.; Aznárez, J.; Greiner, D.; Maeso, O., Shape design optimization of road acoustic barriers featuring top-edge devices by using genetic algorithms and boundary elements, Eng. Anal. Bound. Elem., 63, 49-60 (2016) · Zbl 1403.74241
[36] Zuo, W.; Saitou, K., Multi-material topology optimization using ordered SIMP interpolation, Struct. Multidiscip. Optim., 55, 1-15 (2017)
[37] Long, H.; Hu, Y.; Jin, X.; Yu, H.; Zhu, H., An optimization procedure for spot-welded structures based on SIMP method, Comput. Mater. Sci., 117, 602-607 (2016)
[38] Sigmund, O., Morphology-based black and white filters for topology optimisation. Structural and Multidisciplinary optimisation, Struct. Multidiscip. Optim., 33, 4, 401-424 (2007)
[39] Liu, S.; Li, Q.; Chen, W.; Hu, R.; Tong, L., H-DGTP—a Heaviside-function based directional growth topology parameterization for design optimization of stiffener layout and height of thin-walled structures, Struct. Multidiscip. Optim., 52, 5, 903-913 (2015)
[40] Xu, S.; Cai, Y.; Cheng, G., Volume preserving nonlinear density filter based on heaviside functions, Struct. Multidiscip. Optim., 41, 4, 495-505 (2010) · Zbl 1274.74419
[41] Kai, L.; Hongwei, Z., A modified optimality criterion method for gray elements suppression, J. Comput. Aided Des. Comput. Graph., 22, 12, 2197-2201 (2010)
[42] Shang, L.; Zhao, G., Optimality criteria-based topology optimisation of a bi-material model for acoustic-structural coupled systems, Eng. Optim., 48, 6, 1060-1079 (2016)
[43] Hassani, B.; Khanzadi, M., An isogeometrical approach to structural topology optimization by optimality criteria, Struct. Multidiscip. Optim., 45, 223-233 (2012) · Zbl 1274.74339
[44] Ishizuka, T.; Fujiwara, K., Performance of noise barriers with various edge shapes and acoustical conditions, Appl. Acoust., 65, 2, 125-141 (2004)
[45] Burton, A. J.; Miller, G. F., The application of integral equation methods to the numerical solution of some exterior boundary-value problems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 323, 201-210 (1971) · Zbl 0235.65080
[46] Marburg, S., Six boundary elements per wavelength. Is that enough?, J. Comput. Acoust., 10, 25-51 (2002) · Zbl 1360.76168
[47] Monegato, G., Numerical evaluation of hypersingular integrals, J. Comput. Appl. Math., 50, 9-31 (1994) · Zbl 0818.65016
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.