Resolving high frequency issues via proper orthogonal decomposition based dynamic isogeometric analysis for structures with dissimilar materials. (English) Zbl 1441.74301
Summary: After space discretization employing traditional dynamic isogeometric analysis of structures (composite type) with/without dissimilar materials, the issues that persist include either using numerically non-dissipative time integration algorithms that induces the high frequency participation (oscillations) in solution, or using dissipative algorithms that can dampen the high frequency participation but simultaneously induce significant loss of total energy of the system. To circumvent this dilemma, we instead develop a novel approach via a proper orthogonal decomposition (POD) based dynamic isogeometric methodology/framework that eliminates (significantly reduces) high-frequency oscillations whilst conserving the physics (e.g., total energy) for dynamic analysis of (composite or hybrid) structures comprising of dissimilar materials. This proposed framework and contributions therein are comprised of three phases, namely, (1) it successfully filters the high-frequency part via first simulating the original IGA semi-discretized structure with dissimilar materials for a few time steps using numerically dissipative type integration schemes, and then obtain the reduced IGA system via POD. (2) We then simulate the reduced IGA system (high-frequency oscillations being eliminated) with instead a numerically non-dissipative algorithm to conserve the underlying physics. As consequence, we successfully preserve the physics (energy) associated with the low frequency modes whilst eliminating/reducing the high-frequency oscillations. (3) Three illustrative examples, with/without dissimilar materials demonstrate the advantages of IGA for applications to structures with dissimilar materials over FEM, in particular, in modeling complex geometries and providing more accurate dynamic solutions with less number of degrees of freedom. Furthermore, we show the effectiveness and efficiency of the proposed approach in further advancing the dynamic isogeometric analyses for both linear, and material and/or geometrically nonlinear cases.
MSC:
| 74S99 | Numerical and other methods in solid mechanics |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 74E30 | Composite and mixture properties |
Keywords:
dynamic isogeometric analysis (IGA); physics conserved whist high-frequency oscillation eliminated; proper orthogonal decomposition (POD); numerically dissipative and non-dissipative time integration algorithmsReferences:
| [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, 4135-4195 (2005) · Zbl 1151.74419 |
| [2] | Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), Wiley: Wiley New York · Zbl 1378.65009 |
| [3] | Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2012), Courier Corporation |
| [4] | Simpson, R. N.; Bordas, S. P.A.; Lian, H.; Trevelyan, J., An isogeometric boundary element method for elastostatic analysis: 2D implementation aspects, Comput. Struct., 118, 2-12 (2013) |
| [5] | 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. Eng., 209, 87-100 (2012) · Zbl 1243.74193 |
| [6] | Peng, X.; Atroshchenko, E.; Kerfriden, P.; Bordas, S. P.A., Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth, Comput. Methods Appl. Mech. Eng., 316, 151-185 (2017) · Zbl 1439.74370 |
| [7] | 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 |
| [8] | Heltai, L.; Arroyo, M.; DeSimone, A., Nonsingular isogeometric boundary element method for Stokes flows in 3D, Comput. Methods Appl. Mech. Eng., 268, 514-539 (2014) · Zbl 1295.76022 |
| [9] | Isogeometric Analysis: A New Paradigm in the Numerical Approximation of PDEs: Cetraro, Italy 2012, Vol. 2161 (2016), Springer · Zbl 1355.65004 |
| [10] | Bartezzaghi, A.; Dedè, L.; Quarteroni, A., Isogeometric analysis of high order partial differential equations on surfaces, Comput. Methods Appl. Mech. Eng., 295, 446-469 (2015) · Zbl 1425.65145 |
| [11] | Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Eng., 195, 41-43, 5257-5296 (2006) · Zbl 1119.74024 |
| [12] | Yu, P.; Anitescu, C.; Tomar, S.; Bordas, S. P.A.; Kerfriden, P., Adaptive isogeometric analysis for plate vibrations: An efficient approach of local refinement based on hierarchical a posteriori error estimation, Comput. Methods Appl. Mech. Eng., 342, 251-286 (2018) · Zbl 1440.74164 |
| [13] | Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J. A.; Hughes, T. J.R.; Lipton, M. A.; Scott, S.; Sederberg, T. W., Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Eng., 199, 5-8, 229-263 (2010) · Zbl 1227.74123 |
| [14] | Giannelli, C.; Jüttler, B.; Speleers, H., THB-splines: The truncated basis for hierarchical splines, Comput. Aided Geom. Des., 29, 7, 485-498 (2012) · Zbl 1252.65030 |
| [15] | Atroshchenko, E.; Tomar, S.; Xu, G.; Bordas, S. P.A., Weakening the tight coupling between geometry and simulation in isogeometric analysis: From sub- and super-geometric analysis to Geometry-Independent Field approximaTion (GIFT), Int. J. Numer. Methods Eng., 114, 10, 1131-1159 (2018) · Zbl 07878352 |
| [16] | Beer, G.; Bordas, S. P.A., Isogeometric Methods for Numerical Simulation, Vol. 561 (2015), Springer · Zbl 1317.74004 |
| [17] | Marussig, B.; Zechner, J.; Beer, G.; Fries, T. P., Fast isogeometric boundary element method based on independent field approximation, Comput. Methods Appl. Mech. Eng., 284, 458-488 (2015) · Zbl 1423.74101 |
| [18] | G. Xu, E. Atroshchenko, S.P.A. Bordas, Geometry-independent field approximation for spline-based finite element methods, in: Proceedings of the 11th World Congress in Computational Mechanics, 2014. |
| [19] | Qian, X. P., Full analytical sensitivities in NURBS based isogeometric shape optimization, Comput. Methods Appl. Mech. Eng., 199, 2059-2071 (2010) · Zbl 1231.74352 |
| [20] | Ding, C. S.; Cui, X. Y.; Li, G. Y., Accurate analysis and thickness optimization of tailor rolled blanks based on isogeometric analysis, Struct. Multidiscip. Optim., 54, 4, 871-887 (2016) |
| [21] | Ding, C. S.; Cui, X. Y.; Huang, G. X.; Li, G. Y.; Cai, Y.; Tamma, K. K., A gradient-based shape optimization scheme via isogeometric exact reanalysis, Eng. Comput., 35, 8, 272-2696 (2018) |
| [22] | Benson, D. J.; Bazilevs, Y.; Hsu, M. C.; Hughes, T. J.R., Isogeometric shell analysis: The Reissner-Mindlin shell, Comput. Methods Appl. Mech. Engrg., 199, 276-289 (2010) · Zbl 1227.74107 |
| [23] | Benson, D. J.; Bazilevs, Y.; Hsu, M. C.; Hughes, T. J.R., A large deformation, rotation-free, isogeometric shell, Comput. Methods Appl. Mech. Engrg., 200, 1367-1378 (2011) · Zbl 1228.74077 |
| [24] | Thai, C. H.; Ferreira, A. J.M.; Bordas, S. P.A.; Rabczuk, T.; Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory, Eur. J. Mech. A, 43, 89-108 (2014) · Zbl 1406.74453 |
| [25] | Thai, C. H.; Nguyen-Xuan, H.; Bordas, S. P.A.; Nguyen-Thanh, N.; Rabczuk, T., Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory, Mech. Adv. Mater. Struct., 22, 6, 451-469 (2015) |
| [26] | Nguyen-Xuan, H.; Tran, L. V.; Thai, C. H.; Kulasegaram, S.; Bordas, S. P.A., Isogeometric analysis of functionally graded plates using a refined plate theory, Composites B, 64, 222-234 (2014) |
| [27] | Ding, C. S.; Cui, X. Y.; Deokar, R. R.; Li, G. Y.; Cai, Y.; Tamma, K. K., Modeling and simulation of steady heat transfer analysis with material uncertainty: Generalized \(n\) th order perturbation isogeometric stochastic method, Numer. Heat Transf. A: Appl., 74, 9, 1565-1582 (2018) |
| [28] | Ding, C. S.; Deokar, R. R.; Cui, X. Y.; Li, G. Y.; Cai, Y.; Tamma, K. K., Proper orthogonal decomposition and Monte Carlo based isogeometric stochastic method for material, geometric and force multi-dimensional uncertainties, Comput. Mech., 63, 3, 521-533 (2019) · Zbl 1469.74120 |
| [29] | Ding, C. S.; Hu, X. B.; Cui, X. Y.; Li, G. Y.; Cai, Y.; Tamma, K. K., Isogeometric generalized \(n\) th order perturbation-based stochastic method for exact geometric modelling of (composite) structures: Static and dynamic analysis with random material parameters, Comput. Methods Appl. Mech. Eng., 346, 1002-1024 (2019) · Zbl 1440.74390 |
| [30] | Ding, C. S.; Deokar, R. R.; Ding, Y. J.; Li, G. Y.; Cui, X. Y.; Tamma, K. K.; Bordas, S. P.A., Model order reduction accelerated Monte Carlo stochastic isogeometric method for the analysis of structures with high-dimensional and independent material uncertainties, Comput. Methods Appl. Mech. Eng., 349, 266-284 (2019) · Zbl 1441.74295 |
| [31] | Ding, C. S.; Cui, X. Y.; Huang, G. Y.; Li, G. Y.; Tamma, K. K., Exact and efficient isogeometric reanalysis of accurate shape and boundary modifications, Comput. Methods Appl. Mech. Engrg., 318, 619-635 (2017) · Zbl 1439.65156 |
| [32] | Ding, C. S.; Cui, X. Y.; Deokar, R. R.; Li, G. Y.; Cai, Y.; Tamma, K. K., An isogeometric independent coefficients (IGA-IC) reduced order method for accurate and efficient transient nonlinear heat conduction analysis, Numer. Heat Transf. A: Appl., 73, 10, 667-684 (2018) |
| [33] | Lorenzis, L. D.; Wriggers, P.; Hughes, T. J.R., Isogeometric contact: a review, GAMM-Mitt., 37, 85-123 (2014) · Zbl 1308.74114 |
| [34] | Provatidis, C. G., Precursors of Isogeometric Analysis (2019), Springer International Publishing |
| [35] | Jüttler, B.; Simeon, B., Isogeometric Analysis and Applications (2014), Springer: Springer Germany · Zbl 1337.65003 |
| [36] | Nguyen, V. P.; Anitescu, C.; Bordas, S. P.A.; Rabczuk, T., Isogeometric analysis: An overview and computer implementation aspects, Math. Comput. Simul., 117, 89-116 (2015) · Zbl 1540.65492 |
| [37] | Vuong, A. V.; Heinrich, C.; Simeon, B., ISOGAT: A 2D tutorial MATLAB code for Isogeometric Analysis, Comput. Aided Geom. Des., 27, 8, 644-655 (2010) · Zbl 1205.65319 |
| [38] | de Falco, C.; Reali, A.; Vázquez, R., GeoPDEs: a research tool for isogeometric analysis of PDEs, Adv. Eng. Softw., 42, 12, 1020-1034 (2011) · Zbl 1246.35010 |
| [39] | Milton, G. W., The theory of composites, (The Theory of Composites (2002), Cambridge University Press: Cambridge University Press Cambridge, UK) · Zbl 0993.74002 |
| [40] | Thai, C. H.; Ferreira, A. J.M.; Bordas, S. P.A.; Rabczuk, T.; Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory, Eur. J. Mech. A, 43, 89-108 (2014) · Zbl 1406.74453 |
| [41] | Nguyen, V. P.; Kerfriden, P.; Bordas, S. P.; Rabczuk, T., Isogeometric analysis suitable trivariate NURBS representation of composite panels with a new offset algorithm, Comput.-Aided Des., 55, 49-63 (2014) |
| [42] | Nguyen, N. T.; Hui, D.; Lee, J.; Nguyen-Xuan, H., An efficient computational approach for size-dependent analysis of functionally graded nanoplates, Comput. Methods Appl. Mech. Eng., 297, 191-218 (2015) · Zbl 1423.74550 |
| [43] | Nguyen, H. X.; Nguyen, T. N.; Abdel-Wahab, M.; Bordas, S. P.A.; Nguyen-Xuan, H.; Vo, T. P., A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory, Comput. Methods Appl. Mech. Eng., 313, 904-940 (2017) · Zbl 1439.74453 |
| [44] | Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z., The Finite Element Method: its Basis and Fundamentals (2005), Elsevier · Zbl 1307.74005 |
| [45] | Wriggers, P., Nonlinear Finite Element Methods (2008), Springer Science & Business Media · Zbl 1153.74001 |
| [46] | Belytschko, T.; Liu, W. K.; Moran, B.; Elkhodary, K., Nonlinear Finite Elements for Continua and Structures (2013), John wiley & sons · Zbl 1279.74002 |
| [47] | Shimada, M.; Tamma, K. K., Explicit time integrators and designs for first/second order linear transient systems, Encyclopaedia Therm. Stresses, 3, 1524-1530 (2013) |
| [48] | Shimada, M.; Masuri, S.; Tamma, K. K., A novel design of an isochronous integration [iintegration] framework for first/second order multidisciplinary transient systems, Int. J. Numer. Methods Eng., 102, 3-4, 867-891 (2015) · Zbl 1352.65371 |
| [49] | Tamma, K. K.; Masuri, S. U., A novel extension of GS4-1 time integrator to fluid dynamics type non-linear problems with illustrations to Burger’s equation, Int. J. Numer. Methods Heat Fluid Flow, 26, 1634-1660 (2016) · Zbl 1356.76174 |
| [50] | Masuri, S. U.; Tamma, K. K., (Application of GS4-1 Time Integration Framework to Nonlinear Heat Transfer: Heat Conduction in Medium with Temperature-Dependent Velocity. Application of GS4-1 Time Integration Framework to Nonlinear Heat Transfer: Heat Conduction in Medium with Temperature-Dependent Velocity, Encyclopedia of Thermal Stresses (2014), Springer Netherlands), 191-206 |
| [51] | Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\( \alpha\) method, J. Appl. Mech., 60, 2, 371-375 (1993) · Zbl 0775.73337 |
| [52] | Deokar, R.; Shimada, M.; Lin, C.; Tamma, K. K., On the treatment of high-frequency issues in numerical simulation for dynamic systems by model order reduction via the proper orthogonal decomposition, Comput. Methods Appl. Mech. Eng., 325, 139-154 (2017) · Zbl 1439.74407 |
| [53] | Zhou, X.; Tamma, K. K., Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics, Internat. J. Numer. Methods Engrg., 59, 5, 597-668 (2004) · Zbl 1068.74616 |
| [54] | Pearson, K., LIII. On lines and planes of closest fit to systems of points in space, Lond. Edinb. Dublin Philoso. Mag. J. Sci., 2, 11, 559-572 (1901) · JFM 32.0710.04 |
| [55] | Volkwein, S., (Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling. Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling, Lecture Notes, vol. 4 (2013), University of Konstanz) |
| [56] | Chaturantabut, S.; Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 5, 2737-2764 (2010) · Zbl 1217.65169 |
| [57] | Cazemier, W., Proper Orthogonal Decomposition and Low Dimensional Models for Turbulent Flows (1997), University of Groningen |
| [58] | Rowley, C. W.; Colonius, T.; Murray, R. M., Model reduction for compressible flows using pod and Galerkin projection, Physica D, 189, 1, 115-129 (2004) · Zbl 1098.76602 |
| [59] | Sharkady, M. T.; Cusumano, J. P.; Kimble, B. W., Experimental measurements of dimensionality and spatial coherence in the dynamics of a flexible-beam impact oscillator, Philos. Trans. R. Soc. London A: Phys. Eng. Sci., 347, 1683, 421-438 (1994) |
| [60] | Kappagantu, R.; Feeny, B. F., An optimal modal reduction of a system with frictional excitation, J. Sound Vib., 224, 5, 863-877 (1999) |
| [61] | Rinaldi, M., Reduced basis method for isogeometric analysis: application to structural problems (2015) |
| [62] | Manzoni, A.; Salmoiraghi, F.; Heltai, L., Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils, Comput. Methods Appl. Mech. Eng., 284, 1147-1180 (2015) · Zbl 1423.76181 |
| [63] | Zhu, S.; Dedè, L.; Quarteroni, A., Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation, ESAIM Control Optim. Calc. Var., 51, 4, 1197-1221 (2017) · Zbl 1381.65086 |
| [64] | Zhu, S.; Dedè, L.; Quarteroni, A., Isogeometric analysis and proper orthogonal decomposition for parabolic problems, Numer. Math., 135, 2, 333-370 (2017) · Zbl 1380.65295 |
| [65] | Sirovich, L., Turbulence and the dynamics of coherent structures. I. Coherent structures, Q. Appl. Math., 45, 3, 561-571 (1987) · Zbl 0676.76047 |
| [66] | Barrault, M.; Maday, Y.; Nguyen, N. C.; Patera, A. T., An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339, 9, 667-672 (2004) · Zbl 1061.65118 |
| [67] | S. Chaturantabut, D.C. Sorensen, Discrete empirical interpolation for nonlinear model reduction, in: Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R. China, December, 2009, pp. 16-18. |
| [68] | Kerfriden, P.; Gosselet, P.; Adhikari, S.; Bordas, S. P.A., Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems, Comput. Methods Appl. Mech. Eng., 200, 5-8, 850-866 (2011) · Zbl 1225.74092 |
| [69] | Kerfriden, P.; Gosselet, P.; Adhikari, S.; Bordas, S.; Passieux, J. C., (POD-based Model Order Reduction for the Simulation of Strong Nonlinear Evolutions in Structures: Application to Damage Propagation. POD-based Model Order Reduction for the Simulation of Strong Nonlinear Evolutions in Structures: Application to Damage Propagation, IOP Conference Series: Materials Science and Engineering (2010), IOP Publishing) |
| [70] | Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1970), McGraw-Hill: McGraw-Hill New York · Zbl 0266.73008 |
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