## IG-DRBEM of three-dimensional transient heat conduction problems.(English)Zbl 07371632

Summary: In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

### MSC:

 65-XX Numerical analysis 76-XX Fluid mechanics

BEMECH
Full Text:

### References:

  Hughes, T. J.; 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  Auricchio, F.; Da Veiga, L. B.; Hughes, T. J.R.; Reali, A.; Sangalli, G., Isogeometric collocation methods, Math Models Methods Appl Sci, 20, 11, 2075-2107 (2010) · Zbl 1226.65091  Moosavi, M. R.; Khelil, A., Isogeometric meshless finite volume method in nonlinear elasticity, Acta Mech, 226, 1, 123-135 (2015) · Zbl 1326.74126  Lin, G.; Zhang, Y.; Hu, Z. Q.; Zhong, H., Scaled boundary isogeometric analysis for 2D elastostatics, Sci China-Phys Mech Astron, 57, 2, 286-300 (2014)  Natarajan, S.; Wang, J.; Song, C.; Birk, C., Isogeometric analysis enhanced by the scaled boundary finite element method, Comput Methods Appl Mech Engrg, 283, 733-762 (2015) · Zbl 1425.65174  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, 87-100 (2012) · Zbl 1243.74193  Scott, M. A.; Simpson, R. N.; Evans, J. A.; Lipton, S.; Bordas, S. P.; Hughes, T. J.R.; Sederberg, T. W., Isogeometric boundary element analysis using unstructured T-splines, Comput Methods Appl Mech Engrg, 254, 197-221 (2013) · Zbl 1297.74156  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  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, 12, 1766-1775 (2012) · Zbl 1351.74138  Nguyen, B. H.; Tran, H. D.; 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  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  An, Z.; Yu, T.; Bui, T. Q.; Wang, C.; Trinh, N. A., Implementation of isogeometric boundary element method for 2D steady heat transfer analysis, Adv Eng Softw, 116, 36-49 (2018)  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  Nardini, D.; Brebbia, C. A., A new approach to free vibration using boundary elements, Bound Elem Methods Engrg (1982) · Zbl 0541.73104  Gao, X. W., The radial integration method for evaluation of domain integrals with boundary-only discretization, Eng Anal Bound Elem, 26, 10, 905-916 (2002) · Zbl 1130.74461  Wrobel, L. C.; Brebbia, C. A., The dual reciprocity boundary element formulation for nonlinear diffusion problems, Comput Methods Appl Mech Engrg, 65, 2, 147-164 (1987) · Zbl 0612.76094  Lu, W. Q.; Liu, J.; Zeng, Y., Simulation of the thermal wave propagation in biological tissues by the dual reciprocity boundary element method, Eng Anal Bound Elem, 22, 3, 167-174 (1998) · Zbl 0932.92010  Albuquerque, E. L.; Sollero, P.; Fedelinski, P., Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems, Comput Struct, 81, 17, 1703-1713 (2003)  Yu, B.; Zhou, H. L.; Chen, H. L.; Tong, Y., Precise time-domain expanding dual reciprocity boundary element method for solving transient heat conduction problems, Int J Heat Mass Transfer, 91, 110-118 (2015)  Gomes, G.; Neto, A. M.D.; Bezerra, L. M.; Silva, R., An object-oriented approach to dual reciprocity boundary element method applied to 2D elastoplastic problems, Multi Model Mat Str, 15, 5, 958-974 (2019)  Yu, B.; Cao, G.; Huo, W. D.; Zhou, H. L.; Atroshchenko, E., Isogeometric dual reciprocity boundary element method for solving transient heat conduction problems with heat sources, J Comput Appl Math, 385, Article 113197 pp. (2021) · Zbl 1466.65110  Gong, Y. P.; Dong, C. Y., An isogeometric boundary element method using adaptive integral method for 3D potential problems, J Comput Appl Math, 319, 141-158 (2017) · Zbl 1360.65291  Gao, X. W.; Davies, T. G., Boundary element programming in mechanics (2002), Camberidge University Press  Gong, Y. P.; Dong, C. Y., An isogeometric boundary element method using adaptive integral method for 3D potential problems, J Comput Appl Math, 319, 141-158 (2017) · Zbl 1360.65291  Zhang, Y. M.; Gong, Y. P.; Gao, X. W., Calculation of 2D nearly singular integrals over high-order geometry elements using the sinh transformation, Eng Anal Bound Elem, 60, 144-153 (2015) · Zbl 1403.65244  Xie, G. Z.; Zhang, J. M.; Dong, Y. Q.; Huang, C.; Li, G. Y., An improved exponential transformation for nearly singular boundary element integrals in elasticity problems, Int J Solids Struct, 51, 6, 1322-1329 (2014)  Gu, Y.; Hua, Q.; Chen, W.; Zhang, C., Numerical evaluation of nearly hyper-singular integrals in the boundary element analysis, Comput Struct, 167, 15-23 (2016)  Qin, X. Y.; Zhang, J. M.; Xie, G. Z.; Zhou, F. L.; Li, G. Y., A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element, J Comput Appl Math, 235, 4174-4186 (2011) · Zbl 1219.65031  Telles, J. C.F., A self-adaptive coordinate transformation for efficient numerical evaluation of general boundary element integrals, Internat J Numer Methods Engrg, 24, 959-973 (1987) · Zbl 0622.65014  Karami, G.; Derakhshan, D., An efficient method to evaluate hyper singular and super singular integrals in boundary integral equations analysis, Eng. Anal. Bound. Elem., 23, 4, 317-326 (1999) · Zbl 0940.65139  Niu, Z. R.; Zhou, H. L., A novel boundary integral equation method for linear elasticity-natural boundary integral equation, Acta Mech Solida Sin, 14, 1, 1-10 (2001)  Wang, J.; Tsay, T. K., Analytical evaluation and application of the singularities in boundary element method, Eng Anal Bound Elem, 29, 3, 241-256 (2005) · Zbl 1182.76915  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, 45, 2856-2864 (2010) · Zbl 1231.65236  Hahn, D. W.; Özisik, M. N., Heat conduction (2012), John Wiley & Sons  Nardini, D.; Brebbia, C. A., A new approach to free vibration analysis using boundary elements, Appl Math Model., 7, 3, 157-162 (1983) · Zbl 0545.73078  Gao, X. W.; Davies, T. G., Adaptive integration in elastoplastic boundary element analysis, J Chin Inst Eng, 23, 3, 349-356 (2000)  Partridge, P. W.; Brebbia, C. A.; Wrobel, L. C., The dual reciprocity boundary element method (1992), Computational Mechanics Publications: Computational Mechanics Publications Southampton Boston · Zbl 0758.65071
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.