Parallel Technology for Solving Nonstationary Heat Conduction Problems in Axisymmetric Domains

L. F. Spevak; O. A. Nefedova

doi:10.17804/2410-9908.2021.6.60-71

L. F. Spevak, O. A. Nefedova

PARALLEL TECHNOLOGY FOR SOLVING NONSTATIONARY HEAT CONDUCTION PROBLEMS IN AXISYMMETRIC DOMAINS

DOI: 10.17804/2410-9908.2021.6.60-71

The paper develops a parallel algorithm and program for solving nonstationary heat conduction and diffusion problems in axisymmetric domains with axisymmetric boundary conditions. The numerical solution is based on the boundary element method. In order to optimize and enhance the effectiveness of the computer implementation of the algorithm, the computations are parallelized and the OpenMP application program interface is used. The program is tested by comparing the calculation results with the data of known exact solutions. The calculations confirm the correctness of the numerical solutions and the possibility of full scaling at different numbers of boundary elements according to the number of cores/processors available. The program is applicable to solving axisymmetric heat conduction and diffusion problems and, as a component of a software system, to solving nonlinear problems.

Acknowledgement: The work was performed under a state assignment, state registration number AAAA-A18-118020790140-5.

Keywords: axisymmetric heat conduction problem, boundary element method, parallel computations, OpenMP

References:

Fedotov V.P., Spevak L.F., Nefedova O.A. Parallel algorithms for strength analysis of hydrogenated structures. Programmnye produkty i sistemy, 2012, vol. 99, No. 3, pp. 235–239. (In Russian).
Fedotov V.P., Spevak L.F., Nefedova O.A. A software package designed to solve problems of the potential theory by the boundary element method Programmnye produkty i sistemy, 2014, vol. 108, No. 4, pp. 178–183. DOI: 10.15827/0236-235X.108.178-182. (In Russian).
Fedotov V.P., Spevak L.F. Analytical integration of kernel functions for solving elasticity problems and potential theory by the method of boundary elements. Matematicheskoe modelirovanie, 2007, vol. 19, No. 2, pp. 87–104. (In Russian).
Fedotov V.P., Spevak L.F. One approach to the derivation of exact integration formulae in the boundary element method. Engineering Analysis with Boundary Elements, 2008, vol. 32, No. 10, pp. 883–888. DOI: 10.1016/j.enganabound.2008.03.001.
Fedotov V.P., Spevak L.F., Nefedova O.A. Моделирование процессов упругопластического деформирования модифицированным методом граничных элементов. Programmnye produkty i sistemy, 2013, vol. 4, No. 4, pp. 253–257. (In Russian).
Spevak L.F., Nefedova O.A. Solving a two-dimensional nonlinear heat conduction equation with degeneration by the boundary element method with the application of the dual reciprocity method. AIP Conference Proceedings, 2016, vol. 1785, 040077. DOI: 10.1063/1.4967134.
Spevak L.F., Nefedova O.A. Solving a two-dimensional nonlinear heat conduction equation with nonzero boundary conditions by the boundary element method. AIP Conference Proceedings, 2017, vol. 1915, 040055. DOI: 10.1063/1.5017403.
Kazakov A.L., Nefedova O.A., Spevak L.F. Solution of the Problem of Initiating the Heat Wave for a Nonlinear Heat Conduction Equation Using the Boundary Element Method. Computational Mathematics and Mathematical Physics, 2019, vol. 59, iss. 6, pp. 1015–1029. DOI: 10.1134/S0965542519060083.
Kazakov A., Spevak L., Nefedova O., Lempert A. On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term. Symmetry-Basel, 2020, vol. 12, iss. 6, article 921. DOI: 10.3390/sym12060921.
Kazakov A.L., Spevak L.F., Nefedova O.A. A Numerical Solution to the Two-Dimensional Nonlinear Degenerate Heat Conduction Equation with a Source. AIP Conference Proceedings, 2020, vol. 2315, 040018. DOI: 10.1063/5.0036718.
Spevak L.F., Nefedova O.A. Parallel technology for solving the poisson equation in axisymmetric domains by the boundary element method. AIP Conference Proceedings, 2018, vol. 2053, 030070. DOI: 10.1063/1.5084431.
Nefedova O.A., Spevak L.F. Parallel Technology for Solving Axisymmetric Problems of the Theory of Elasticity by the Boundary Element Method. AIP Conference Proceedings, 2020, vol. 2315, 020030. DOI: 10.1063/5.0037021.
Rizzo F.J., Shippy D.J. A method of solution for certain problems of transient heat conduction. AIAA J., 1970, vol. 8, pp. 2004–2009. DOI:10.2514/3.6038.
Shaw R.P. An integral equation approach to diffusion. International Journal of Heat and Mass Transfer, 1974, vol. 17 (6), pp. 693–699. DOI: 10.1016/0017-9310(74)90202-6.
Brebbia C.A., Walker S. Boundary Element Techniques in Engineering, Newnes–Butterworths, London, 1980. ISBN: 9781483102566.
Wrobel L.C., Brebbia C.A. A formulation of the boundary element method for axisymmetric transient heat conduction. International Journal of Heat and Mass Transfer, 1981, vol. 24, pp. 843–850. DOI: 10.1016/S0017-9310(81)80007-5.
Zhu S.P. Time-dependent reaction diffusion problems and the LTDRM approach. In: M. Goldberg, ed. Boundary Integral Methods: Numerical and Mathematical Aspects. Computational Mechanics Publications, Southampton, Boston, 1999, pp. 1–35.
Sutradhar A, Paulino G.H, Gray L.J. Transient heat conduction in homogeneous and nonhomogeneous materials by the Laplace Transform Galerkin boundary element method. Engineering Analysis with Boundary Elements, vol. 26 (2), pp. 119–132. DOI: 10.1016/S0955-7997(01)00090-X.
Brebbia C.A., Telles J.F.C., Wrobel L.C. Boundary Element Techniques, Springer-Verlag, Berlin, Neidelberg, New-York, Tokyo, 1984, 466 р. ISBN: 978-3-642-48862-7 (print), 978-3-642-48860-3 (online). DOI: 10.1007/978-3-642-48860-3.
Abramowitz M., Stegun I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. In: M. Abramowitz, I. A. Stegun, eds. Dover Books on Advanced Mathematics, Dover Publications, New York, 1965.
Kronrod A.S. Uzly i vesa kvadraturnykh formul [Nodes and weights of quadrature formulas]. Moscow, Nauka Publ., 1964, 143 p. (In Russian).
Lifanov I.K. Metod singulyarnykh integralnykh uravnenii i chislennyi eksperiment [Singular integral equations method and numerical experiment]. Moscow, Yanus Publ.,1995, 520 p. ISBN 5-88929-003-7. (In Russian).
GSL–GNU Scientific Library. Available at: http://www.gnu.org/software/gsl/ (accessed 07.04.2021).
Boost C++ Libraries. Available at: http://www.boost.org/ (accessed 07.04.2021).
What is OpenMP? PARALLEL.RU. Available at: https://parallel.ru/tech/tech_dev/openmp.html (accessed 11.05.2021).
OpenMP. Available at: http://www.openmp.org/ (accessed 11.05.2021).
Lykov A.V. Teoriya teploprovodnosti [Heat conduction theory]. Moscow, Vysshaya shkola Publ., 1967, 597 p. (In Russian).

Л. Ф. Спевак, О. А. Нефедова

ПАРАЛЛЕЛЬНАЯ ТЕХНОЛОГИЯ РЕШЕНИЯ НЕСТАЦИОНАРНЫХ ЗАДАЧ ТЕПЛОПРОВОДНОСТИ В ОСЕСИММЕТРИЧНОЙ ПОСТАНОВКЕ

Работа посвящена разработке параллельного алгоритма и программы для решения нестационарных задач теплопроводности и диффузии в осесимметричных областях при осесимметричных граничных условиях. В основе численного решения лежит метод граничных элементов. Для оптимизации и повышения эффективности компьютерной реализации алгоритма было выполнено распараллеливание вычислений и привлечен открытый стандарт параллельного программирования OpenMP. Разработанная программа была протестирована сравнением результатов расчетов с данными известных точных решений. Расчеты подтверждают корректность численных решений и возможность полного масштабирования при различных количествах граничных элементов в соответствии с количеством доступных ядер/процессоров. Программа может быть использована для решения осесимметричных задач теплопроводности и диффузии, а также как составляющая программного комплекса для решения нелинейных задач.

Благодарность: Работа выполнена в рамках государственного задания, номер государственной регистрации АААА-А18-118020790140-5.

Ключевые слова: осесимметричная задача теплопроводности, метод граничных элементов, параллельные вычисления, OpenMP

Библиография:

Федотов В. П., Спевак Л. Ф., Нефедова О. А. Параллельные алгоритмы для анализа прочности наводороженных конструкций // Программные продукты и системы. – 2012. – T. 99, № 3. – C. 235–239.
Федотов В. П., Спевак Л. Ф., Нефедова О. А. Программный комплекс для решения задач теории потенциала методом граничных элементов // Программные продукты и системы. – 2014. – T. 108, № 4. – C. 178–183. – DOI: 10.15827/0236-235X.108.178-182.
Федотов В. П., Спевак Л. Ф. Аналитическое интегрирование функций влияния для решения задач упругости и теории потенциала методом граничных элементов // Математическое моделирование. – 2007. – Т. 19, № 2. – С. 87–104.
Fedotov V. P., Spevak L. F. One approach to the derivation of exact integration formulae in the boundary element method // Engineering Analysis with Boundary Elements. – 2008. – Vol. 32, No. 10. – P. 883–888. – DOI: 10.1016/j.enganabound.2008.03.001.
Федотов В. П., Спевак Л. Ф., Нефедова О. А. Моделирование процессов упругопластического деформирования модифицированным методом граничных элементов // Программные продукты и системы. – 2013. – T. 4, № 4. – С. 253–257.
Spevak L. F., Nefedova O. A. Solving a two-dimensional nonlinear heat conduction equation with degeneration by the boundary element method with the application of the dual reciprocity method // AIP Conference Proceedings. – 2016. – Vol. 1785. – P. 040077. – DOI: 10.1063/1.4967134.
Spevak L. F., Nefedova O. A. Solving a two-dimensional nonlinear heat conduction equation with nonzero boundary conditions by the boundary element method // AIP Conference Proceedings. – 2017. – Vol. 1915. – P. 040055. – DOI: 10.1063/1.5017403.
Kazakov A. L., Nefedova O. A., Spevak L. F. Solution of the Problem of Initiating the Heat Wave for a Nonlinear Heat Conduction Equation Using the Boundary Element Method // Computational Mathematics and Mathematical Physics. – 2019. – Vol. 59, iss. 6. – P. 1015–1029. – DOI: 10.1134/S0965542519060083.
On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term / A. Kazakov, L. Spevak, O. Nefedova, A. Lempert // Symmetry. – 2020. – Vol. 12, iss. 6. – article 921. – DOI: 10.3390/sym12060921.
Kazakov A. L., Spevak L. F., Nefedova O. A. A Numerical Solution to the Two-Dimensional Nonlinear Degenerate Heat Conduction Equation with a Source // AIP Conference Proceedings. – 2020. – Vol. 2315. – P. 040018. – DOI: 10.1063/5.0036718.
Spevak L. F., Nefedova O. A. Parallel technology for solving the poisson equation in axisymmetric domains by the boundary element method // AIP Conference Proceedings. – 2018. – Vol. 2053. – P. 030070. – DOI: 10.1063/1.5084431.
Nefedova O. A., Spevak L. F. Parallel Technology for Solving Axisymmetric Problems of the Theory of Elasticity by the Boundary Element Method // AIP Conference Proceedings. – 2020. – Vol. 2315. – P. 020030. – DOI: 10.1063/5.0037021.
Rizzo F. J., Shippv, D. J. A method of solution for certain problems of transient heat conduction // IAA J. – 1970. – Vol. 8. – P. 2004–2009. – DOI:10.2514/3.6038.
Shaw R. P. An integral equation approach to diffusion // International Journal of Heat and Mass Transfer. – 1974. – Vol. 17. – P. 693–699. – DOI: 10.1016/0017-9310(74)90202-6.
Brebbia C. A., Walker S. Boundary Element Techniques in Engineering. – London : Newnes–Butterworths, 1980. – ISBN: 9781483102566.
Wrobel L. C., Brebbia C. A. A formulation of the boundary element method for axisymmetric transient heat conduction // International Journal of Heat and Mass Transfer. – 1981. – Vol. 24. – P. 843–850. – DOI: 10.1016/S0017-9310(81)80007-5.
Zhu S. P. Time-dependent reaction diffusion problems and the LTDRM approach // Boundary Integral Methods: Numerical and Mathematical Aspects. Southampton / ed. by M. Goldberg. – Boston : Computational Mechanics Publications, 1999. – P. 1–35.
Sutradhar A., Paulino G. H, Gray L. J. Transient heat conduction in homogeneous and nonhomogeneous materials by the Laplace Transform Galerkin boundary element method // Engineering Analysis with Boundary Elements. – 2002. – Vol. 26. – P. 119–132. – DOI: 10.1016/S0955-7997(01)00090-X.
Brebbia C. A., Telles J. F. C., Wrobel L. C. Boundary Element Techniques. – Berlin, Nei-delberg, New-York, Tokyo : Springer-Verlag, 1984. – 466 р. – ISBN 978-3-642-48862-7. – DOI: 10.1007/978-3-642-48860-3.
Abramowitz M., Stegun I. A. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables // Dover Books on Advanced Mathematics / ed. by M. Abramowitz, I. A. Stegun. – New York : Dover Publications, 1965.
Кронрод А. С. Узлы и веса квадратурных формул. – М. : Наука, 1964. – 143 с.
Лифанов И. К. Метод сингулярных интегральных уравнений и численный эксперимент. – М. : Янус, 1995. – 520 с. – ISBN 5-88929-003-7.
GSL–GNU Scientific Library. Available at: http://www.gnu.org/software/gsl/ (accessed 07.04.2021).
Boost C++ Libraries. Available at: http://www.boost.org/ (accessed 07.04.2021).
What is OpenMP? PARALLEL.RU. Available at: https://parallel.ru/tech/tech_dev/openmp.html (accessed 11.05.2021).
OpenMP. Available at: http://www.openmp.org/ (accessed 11.05.2021).
Лыков А. В. Теория теплопроводности. – М. : Высшая школа, 1967. – 597 с.

Библиографическая ссылка на статью

Spevak L. F., Nefedova O. A. Parallel Technology for Solving Nonstationary Heat Conduction Problems in Axisymmetric Domains // Diagnostics, Resource and Mechanics of materials and structures. - 2021. - Iss. 5. - P. 60-71. -
DOI: 10.17804/2410-9908.2021.6.60-71. -
URL: http://dream-journal.org/issues/2021-5/2021-5_349.html
(accessed: 21.12.2024).