Parallelizing the Solution of the Nonlinear Heat Conduction Problem with the Application of the Opencl Library

L. F. Spevak; O. A. Nefedova

doi:10.17804/2410-9908.2016.6.080-091

L. F. Spevak, O. A. Nefedova

PARALLELIZING THE SOLUTION OF THE NONLINEAR HEAT CONDUCTION PROBLEM WITH THE APPLICATION OF THE OPENCL LIBRARY

DOI: 10.17804/2410-9908.2016.6.080-091

The paper deals with the application of parallel computation methods to the numerical solution of the nonlinear boundary value problem for the degenerate two-dimensional differential heat conduction equation. The nonlinearity of the problem stems from the power dependence of the thermal conductivity coefficient on temperature. The solution algorithm is based on the boundary element method with the application of the dual reciprocity method enabling all the computations to be brought to the boundary of the problem solution domain. A program has been developed from the presented computational algorithm. To accelerate the computation as much as possible, we use parallel programming processes and graphics processors. The program is written in the С++ programming language with the use of the OpenMP and OpenCL open standards. An example is considered to illustrate the work of the algorithm and the program; the calculation accuracy and the calculation speed are analyzed.

Keywords: parallel computation, OpenMP, OpenCL, nonlinear heat conduction problem, boundary element method, analytical integration, radial basis functions

References:

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Л. Ф. Спевак, О. А. Нефедова

РАСПАРАЛЛЕЛИВАНИЕ РЕШЕНИЯ НЕЛИНЕЙНОЙ ЗАДАЧИ ТЕПЛОПРОВОДНОСТИ С ИСПОЛЬЗОВАНИЕМ БИБЛИОТЕКИ OPENCL

Статья посвящена применению методов параллельных вычислений к численному решению нелинейной краевой задачи для двумерного дифференциального уравнения теплопроводности с вырождением. Нелинейность задачи обусловлена степенной зависимостью коэффициента теплопроводности от температуры. Алгоритм решения основан на методе граничных элементов с использованием метода двойственной взаимности, позволяющего свести все вычисления на границу области решения задачи. На основе представленного вычислительного алгоритма разработана программа. Для максимального ускорения счета задействованы технологии параллельного программирования и графические процессоры. Программа написана на языке программирования С++ с использованием открытых стандартов OpenMP и OpenCL. Рассмотрен пример, иллюстрирующий работу алгоритма и программы, выполнен анализ точности и скорости счета.

Ключевые слова: параллельные вычисления, OpenMP, OpenCL, нелинейная задача теплопроводности, метод граничных элементов, аналитическое интегрирование, радиальные базисные функции

Библиография:

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Petaccia G., Leporati F., Torti E. OpenMP and CUDA simulations of Sella Zerbino Dam break on unstructured grids // Computational Geosciences. – 2016. – Vol. 20, no. 10. – P. 1123–1132. – DOI: 10.1007/s10596-016-9580-5.
Lung diaphragm tracking in CBCT images using spatio–temporal MRF / M. Sundarapandian, R. Kalpathi, R. A. C. Siochi, A. S. Kadam // Computerized Medical Imaging and Graphics. – 2016. – Vol. 53. – P. 9–18. – DOI: 10.1016/j.compmedimag.2016.07.001.
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Local gas holdup simulation and validation of industrial-scale aerated bioreactors / C. Witz, D. Treffer, T. Hardiman, J. Khinast // Chemical Engineering Science. – 2016. – Vol. 152. – P. 636–648. – DOI: 10.1016/j.ces.2016.06.053.
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CPU–GPU mixed implementation of virtual node method for real–time interactive cutting of deformable objects using OpenCL / S. Y. Jia, W. Z. Zhang, X. K. Yu, Z. K. Pan // International Journal of Computer Assisted Radiology and Surgery. – 2015. – Vol. 10, no. 9. – P. 1477–1491. – DOI: 10.1007/s11548-014-1147-0.
CUDA Zone. NVIDIA Developer. – URL: https://developer.nvidia.com/cuda–zone
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OpenMP. – URL: http://www.openmp.org/
Parallel Programming in OpenMP / R. Chandra, L. Dagum, D. Kohr, D. Maydan, J. McDonald, R. Melon. – San-Francisco : Morgan Kaufmann Publishers, 2001. – 231 p. – ISBN 1–55860–671–8.
OpenACC. Directives for Accelerators. – URL: http://www.openacc.org/
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.
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Kazakov A. L., Spevak L. F. Numerical and analytical studies of a nonlinear parabolic equation with boundary conditions of a special form // Applied Mathematical Modelling. – 2013. – Vol. 37, iss. 10–11. – P. 6918–6928. – DOI: 10.1016/j.apm.2013.02.026.
Спевак Л. Ф., Нефедова О. А. Решение нелинейного уравнения теплопроводности методом граничных элементов с использованием метода двойственной взаимности // Международный журнал прикладных и фундаментальных исследований. – 2015. – № 12–1. – С. 50–55. – URL: http://www.applied–research.ru/ru/article/view?id=7813
Kazakov A. L., Spevak L. F. An analytical and numerical study of a nonlinear parabolic equation with degeneration for the cases of circular and spherical symmetry // Applied Mathematical Modelling. – 2015. – Vol. 40, iss. 2. – P. 1333–1343. – DOI: 10.1016/j.apm.2015.06.038.
The Porous Medium Equation: Mathematical Theory / edited by J. L. Vazquez. – Oxford : Clarendon Press, 2006. – 648 p. – ISBN 978-0-19-856903-9. – DOI: 10.1093/acprof:oso/9780198569039.001.0001.
Brebbia C. A., Telles J. F. C., Wrobel L. C. Boundary Element Techniques. – Berlin, Neidelberg, New-York, Tokyo : Springer-Verlag, 1984. – 466 p. – ISBN 978-3-642-48862-7. – DOI: 10.1007/978-3-642-48860-3..
Nardini D., Brebbia C. A. A New Approach to Free Vibration Analysis using Boundary Elements // Applied Mathematical Modelling. – 1983. – Vol. 7, no. 3. – P. 157–162. – DOI: 10.1016/0307-904X(83)90003-3.
GSL-GNU Scientific Library. – URL: http://www.gnu.org/software/gsl/
Boost C++ Libraries. – URL: http://www.boost.org/

Библиографическая ссылка на статью

Spevak L. F., Nefedova O. A. Parallelizing the Solution of the Nonlinear Heat Conduction Problem with the Application of the Opencl Library // Diagnostics, Resource and Mechanics of materials and structures. - 2016. - Iss. 6. - P. 80-91. -
DOI: 10.17804/2410-9908.2016.6.080-091. -
URL: http://dream-journal.org/issues/content/article_113.html
(accessed: 21.12.2024).