RI-IGABEM for 2D viscoelastic problems and its application to solid propellant grains. (English) Zbl 1506.74469
Summary: The isogeometric boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation, excellent field approximation and only boundary discretization property. In this paper, IGABEM based on radial integration method (RI-IGABEM) is used for viscoelastic analysis of solid propellant grain. The memory stress, as the initial stress, leads to the boundary-domain integral equations and thus eliminates the only boundary discretization advantage of boundary element method (BEM). The radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral by means of the applied points. The usage of RIM makes it possible to only store the strains on the applied points. Meanwhile, Prony-series is used to discretize the general integrals and to store the two most recent time-step strains rather than the time-step strains of the entire process. The combination between RIM and Prony-series will help reduce the storage space and computational time. In addition, by using the fundamental solutions for linear elastic problems and the regularized technologies, the singular integrals can be solved through the previous methods, such as the Telles scheme and element sub-division method. In order to validate the accuracy and robustness of RI-IGABEM in viscoelastic analysis, the influence of the number and position of applied points as well as the time interval on viscoelastic analysis is discussed through comparing with cell discretization methods. A set of numerical examples demonstrates the ability of the scheme to simulate the viscoelastic problems.
MSC:
| 74S15 | Boundary element methods applied to problems in solid mechanics |
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 65D07 | Numerical computation using splines |
| 65D12 | Numerical radial basis function approximation |
| 74B05 | Classical linear elasticity |
| 74D05 | Linear constitutive equations for materials with memory |
Keywords:
isogeometric boundary element method; viscoelastic analysis; radial integration method; prony series; solid propellant grainSoftware:
DistMeshReferences:
| [1] | Simões Hoffmann, L. F.; Parquet Bizarria, F. C.; Parquet Bizarria, J. W., Detection of liner surface defects in solid rocket motors using multilayer perceptron neural networks, Polym. Test., 88, Article 106559 pp. (2020) |
| [2] | Marimuthu, R.; Nageswara Rao, B., Development of efficient finite elements for structural integrity analysis of solid rocket motor propellant grains, Int. J. Press. Vessels Pip., 111-112, 131-145 (2013) |
| [3] | Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z., The Standard Discrete System and Origins of the Finite Element Method (2013), Butterworth-Heinemann: Butterworth-Heinemann Oxford · Zbl 1307.74005 |
| [4] | Cruse, T. A., A direct formulation and numerical solution of the general transient elastodynamic problem. II, J. Math. Anal. Appl., 22, 341-355 (1968) · Zbl 0167.16302 |
| [5] | Cruse, T. A., Numerical solutions in three dimensional elastostatics, Int. J. Solids Struct., 5, 1259-1274 (1969) · Zbl 0181.52404 |
| [6] | Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), Wiley: Wiley UK · Zbl 1378.65009 |
| [7] | Marussig, B.; Zechner, J.; Beer, G.; Fries, T.-P., Fast isogeometric boundary element method based on independent field approximation, Comput. Methods Appl. Mech. Engrg., 284, 458-488 (2015) · Zbl 1423.74101 |
| [8] | Beer, G.; Marussig, B.; Duenser, C., The Isogeometric Boundary Element Method (2020), Springer · Zbl 1422.65001 |
| [9] | Beer, G.; Duenser, C., Isogeometric boundary element analysis of problems in potential flow, Comput. Methods Appl. Mech. Engrg., 347, 517-532 (2019) · Zbl 1440.76102 |
| [10] | 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 |
| [11] | Lian, H.; Kerfriden, P.; Bordas, S. P.A., Implementation of regularized isogeometric boundary element methods for gradient-based shape optimization in two-dimensional linear elasticity, Internat. J. Numer. Methods Engrg., 106, 972-1017 (2016) · Zbl 1352.74467 |
| [12] | Liu, C.; Chen, L. L.; Zhao, W. C.; Chen, H. B., Shape optimization of sound barrier using an isogeometric fast multipole boundary element method in two dimensions, Eng. Anal. Bound. Elem., 85, 142-157 (2017) · Zbl 1403.76092 |
| [13] | Chen, L. L.; Lian, H.; Liu, Z.; Chen, H. B.; Atroshchenko, E.; Bordas, S. P.A., Structural shape optimization of three dimensional acoustic problems with isogeometric boundary element methods, Comput. Methods Appl. Mech. Engrg., 355, 926-951 (2019) · Zbl 1441.74290 |
| [14] | Xu, C.; Dong, C.; Dai, R., RI-IGABEM based on PIM in transient heat conduction problems of FGMs, Comput. Methods Appl. Mech. Engrg., 374, Article 113601 pp. (2021) · Zbl 1506.74472 |
| [15] | 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. Engrg., 316, 151-185 (2017) · Zbl 1439.74370 |
| [16] | Peake, M. J.; Trevelyan, J.; Coates, G., Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems, Comput. Methods Appl. Mech. Engrg., 259, 93-102 (2013) · Zbl 1286.65176 |
| [17] | Yoon, M.; Lee, J.; Koo, B., Shape design optimization of thermoelasticity problems using isogeometric boundary element method, Adv. Eng. Softw., 149, Article 102871 pp. (2020) |
| [18] | Sun, D.; Dong, C., Shape optimization of heterogeneous materials based on isogeometric boundary element method, Comput. Methods Appl. Mech. Engrg., 370, Article 113279 pp. (2020) · Zbl 1506.74322 |
| [19] | Hu, Q.; Chouly, F.; Hu, P.; Cheng, G.; Bordas, S. P.A., Skew-symmetric Nitsche’s formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact, Comput. Methods Appl. Mech. Engrg., 341, 188-220 (2018) · Zbl 1440.74403 |
| [20] | 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, 1163-1182 (2014) · Zbl 1398.74379 |
| [21] | Peake, M. J.; 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 |
| [22] | Kostas, K. V.; Ginnis, A. I.; Politis, C. G.; Kaklis, P. D., Ship-hull shape optimization with a T-spline based BEM-isogeometric solver, Comput. Methods Appl. Mech. Engrg., 284, 611-622 (2015) · Zbl 1425.65201 |
| [23] | Coox, L.; Greco, F.; Atak, O.; Vandepitte, D.; Desmet, W., A robust patch coupling method for NURBS-based isogeometric analysis of non-conforming multipatch surfaces, Comput. Methods Appl. Mech. Engrg., 316, 235-260 (2017) · Zbl 1439.65154 |
| [24] | Ginnis, A. I.; Kostas, K. V.; Politis, C. G.; Kaklis, P. D.; Belibassakis, K. A.; Gerostathis, T. P.; Scott, M. A.; Hughes, T. J.R., Isogeometric boundary-element analysis for the wave-resistance problem using T-splines, Comput. Methods Appl. Mech. Engrg., 279, 425-439 (2014) · Zbl 1423.74270 |
| [25] | Wang, Y. J.; Benson, D. J.; Nagy, A. P., A multi-patch nonsingular isogeometric boundary element method using trimmed elements, Comput. Mech., 56, 173-191 (2015) · Zbl 1329.65281 |
| [26] | Nguyen, V. P.; Anitescu, C.; Bordas, S. P.A.; Rabczuk, T., Isogeometric analysis: An overview and computer implementation aspects, Math. Comput. Simulation, 117, 89-116 (2015) · Zbl 1540.65492 |
| [27] | Yadav, A.; Godara, R. K.; Bhardwaj, G., A review on XIGA method for computational fracture mechanics applications, Eng. Fract. Mech., 230, Article 107001 pp. (2020) |
| [28] | Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. Comput. Phys., 73, 325-348 (1987) · Zbl 0629.65005 |
| [29] | Zheng, C. J.; Bi, C. X.; Zhang, C. Z.; Zhang, Y. B.; Chen, H. B., Fictitious eigenfrequencies in the BEM for interior acoustic problems, Eng. Anal. Bound. Elem., 104, 170-182 (2019) · Zbl 1464.76120 |
| [30] | Bebendorf, M., Approximation of boundary element matrices, Numer. Math., 86, 565-589 (2000) · Zbl 0966.65094 |
| [31] | Beylkin, G.; Coifman, R.; Rokhlin, V., Fast wavelet transforms and numerical algorithms I, Commun. Pure Appl. Math. Comput., 44, 141-183 (2010) · Zbl 0722.65022 |
| [32] | Phillips, J. R.; White, J. K., A Precorrected-FFT method for electrostatic analysis of complicated 3-d structures, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 16, 1059-1072 (1997) |
| [33] | Lorenzana, A.; Garrido, J. A., Analysis of the elastic-plastic problem involving finite plastic strain using the boundary element method, Comput. Struct., 73, 147-159 (1999) · Zbl 1048.74599 |
| [34] | Gun, H., Elasto-plastic static stress analysis of 3D contact problems with friction by using the boundary element method, Eng. Anal. Bound. Elem., 28, 779-790 (2004) · Zbl 1130.74444 |
| [35] | Dong, C. Y.; Yao, Z. H.; Du, Q. H., Application of boundary element method in 2-D elasro-plastic contact problem, (Lee, W. B., Advances in Engineering Plasticity and Its Applications (1993), Elsevier: Elsevier Oxford), 653-658 |
| [36] | Gao, X.-W.; Zheng, B.-J.; Yang, K.; Zhang, C., Radial integration BEM for dynamic coupled thermoelastic analysis under thermal shock loading, Comput. Struct., 158, 140-147 (2015) |
| [37] | Telles, J. C.F., A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Internat. J. Numer. Methods Engrg., 24, 959-973 (1987) · Zbl 0622.65014 |
| [38] | 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, 421-431 (1998) · Zbl 0941.74075 |
| [39] | Liu, Y.; Rudolphi, T. J., Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations, Eng. Anal. Bound. Elem., 8, 301-311 (1991) |
| [40] | Liu, Y. J.; Rudolphi, T. J., New identities for fundamental solutions and their applications to non-singular boundary element formulations, Comput. Mech., 24, 286-292 (1999) · Zbl 0969.74073 |
| [41] | Liu, Y. J., On the simple-solution method and non-singular nature of the BIE/BEM — a review and some new results, Eng. Anal. Bound. Elem., 24, 789-795 (2000) · Zbl 0974.65110 |
| [42] | Ma, L.; Korsunsky, A. M., A note on the Gauss-Jacobi quadrature formulae for singular integral equations of the second kind, Int. J. Fract., 126, 399-405 (2004) · Zbl 1187.65138 |
| [43] | Huber, O.; Lang, A.; Kuhn, G., Evaluation of the stress tensor in 3D elastostatics by direct solving of hypersingular integrals, Internat. J. Numer. Methods Engrg., 39, 2555-2573 (1993) · Zbl 0880.73072 |
| [44] | Karami, G.; Derakhshan, D., An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis, Eng. Anal. Bound. Elem., 23, 317-326 (1999) · Zbl 0940.65139 |
| [45] | Cerrolaza, M.; Alarcon, E., A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods, Internat. J. Numer. Methods Engrg., 28, 987-999 (1989) · Zbl 0679.73040 |
| [46] | Niu, Z.; Wang, X.; Zhou, H., Non-singular algorithm for the evaluation of the nearly singular integrals in boundary element methods, Chinese J. Appl. Mech., 18, 1-8 (2001) |
| [47] | Kutt, H. R., The numerical evaluation of principal value integrals by finite-part integration, Numer. Math., 24, 205-210 (1975) · Zbl 0306.65010 |
| [48] | Gao, X. W., An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals, Comput. Methods Appl. Mech. Engrg., 199, 2856-2864 (2010) · Zbl 1231.65236 |
| [49] | Gong, Y. P.; Dong, C. Y.; Qin, X. C., An isogeometric boundary element method for three dimensional potential problems, J. Comput. Appl. Math., 313, 454-468 (2017) · Zbl 1353.65129 |
| [50] | 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 |
| [51] | Beer, G.; Marussig, B.; Zechner, J.; Dünser, C.; Fries, T.-P., Isogeometric boundary element analysis with elasto-plastic inclusions. Part 1: Plane problems, Comput. Methods Appl. Mech. Engrg., 308, 552-570 (2016) · Zbl 1439.74057 |
| [52] | Beer, G.; Mallardo, V.; Ruocco, E.; Marussig, B.; Zechner, J.; Dünser, C.; Fries, T.-P., Isogeometric boundary element analysis with elasto-plastic inclusions. Part 2: 3-D problems, Comput. Methods Appl. Mech. Engrg., 315, 418-433 (2017) · Zbl 1439.74056 |
| [53] | Fung, Y. C.; Drucker, D., Foundation of Solid Mechanics (1965), Prentiee-Hall: Prentiee-Hall New Jersey |
| [54] | Lee, E. H., Stress analysis for linear viscoelastic materials, Rheol. Acta, 1, 426-430 (1961) |
| [55] | Williams, M. L.; Landel, R. F.; Ferry, J. D., The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids, J. Am. Chem. Soc., 77, 3701-3707 (1955) |
| [56] | Huang, Q.; Chen, J.; Qiang, H.; Wang, X.; Yue, C.; Li, A., Boundary element method for local stress field analysis of inclusions in solid propellant grain, J. Soild Rocket Technol., 42, 275-307 (2019) |
| [57] | Gong, Y. P.; Yang, H. S.; Dong, C. Y., A novel interface integral formulation for 3D steady state thermal conduction problem for a medium with non-homogenous inclusions, Comput. Mech., 63, 181-199 (2019) · Zbl 1464.74047 |
| [58] | Banerjee, P. K.; Raveendra, S. T., Adcanced boundary element analysis of two- and three-dimensional problems of elasto-plasticity, Intern. J. Numer. Methods Eng., 23, 985-1002 (1986) · Zbl 0592.73040 |
| [59] | Feng, W.-Z.; Gao, L.-F.; Du, J.-M.; Qian, W.; Gao, X.-W., A meshless interface integral BEM for solving heat conduction in multi-non-homogeneous media with multiple heat sources, Int. Commun. Heat Mass Transfer, 104, 70-82 (2019) |
| [60] | Marques, S. P.C.; Creus, G. J., Computational Viscoelasticity (2012), London |
| [61] | Wang, Y. J.; Benson, D. J., Multi-patch nonsingular isogeometric boundary element analysis in 3D, Comput. Methods Appl. Mech. Engrg., 293, 71-91 (2015) · Zbl 1425.65203 |
| [62] | E.C. Ting, Stress analysis for linear viscoelastic cylinders, 8 (1970) 18-22. · Zbl 0185.53601 |
| [63] | P.-O. Persson, G. Strang, A simple mesh generator in MATLAB, 46 (2004) 329-345. · Zbl 1061.65134 |
| [64] | R.L. Taylor, K.S. Pister, G.L. Goudreau, Thermomechanical analysis of viscoelastic solids, 2 (1970) 45-59. · Zbl 0252.73006 |
| [65] | Altenbach, J.; Mechanic, Z., Book review: Martin H. Sadd Elasticity - Theory, applications, and numerics, J. Appl. Math., 85, 907-908 (2010) |
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.
