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L. F. Spevak, O. A. Nefedova


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


  1. Aguilar–Leal O., Fuentes–Aguilar R.Q., Chairez I., Garcia–Gonzalez A., Huegel J.C. Distributed parameter system identification using finite element differential neural networks. Applied Soft Computing, 2016, vol. 43, pp. 633–642. DOI: 10.1016/j.asoc.2016.01.004.
  2. 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, pp. 1123–1132. DOI: 10.1007/s10596-016-9580-5.
  3. Sundarapandian M., Kalpathi R., Siochi R.A.C., Kadam A.S. Lung diaphragm tracking in CBCT images using spatio-temporal MRF. Computerized Medical Imaging and Graphics, 2016, vol. 53, pp. 9–18. DOI: 10.1016/j.compmedimag.2016.07.001.
  4. Li K.L., Yang W.D., Li K.Q. A Hybrid Parallel Solving Algorithm on GPU for Quasi–Tridiagonal System of Linear Equations. IEEE Transactions on Parallel and Distributed Systems, 2016, vol. 27, no. 10, pp. 2795–2808. DOI: 10.1109/TPDS.2016.2516988.
  5. Witz C., Treffer D., Hardiman T., Khinast J. Local gas holdup simulation and validation of industrial–scale aerated bioreactors. Chemical Engineering Science, 2016, vol. 152, pp. 636–648. DOI: 10.1016/j.ces.2016.06.053.
  6. Tredak P., Rudnicki W.R., Majewski J.A. Efficient implementation of the many-body Reactive Bond Order (REBO) potential on GPU. Journal of Computational Physics, 2016, vol. 321, pp. 556–570. DOI: 10.1016/j.jcp.2016.05.061.
  7. Jia S.Y., Zhang W.Z., Yu X.K., Pan Z.K. CPU-GPU mixed implementation of virtual node method for real-time interactive cutting of deformable objects using OpenCL. International Journal of Computer Assisted Radiology and Surgery, 2015, vol. 10, no. 9, pp. 1477–1491. DOI: 10.1007/s11548-014-1147-0.
  8. CUDA Zone. NVIDIA Developer. Available at: https://developer.nvidia.com/cuda–zone (accessed 12.06.2016).
  9. OpenCL The Open Standard for Parallel Programming of Heterogeneous Systems. Available at: https://www.khronos.org/opencl (accessed 05.06.2015).
  10. Munshi A., Gaster B.R., Mattson T.G., Fung J., Ginsburg D. OpenCL Programming Guide, Addison-Wesley, Upper Saddle River, NJ, Boston, Indianapolis, San Francisco, New York, Toronto, Montreal, London, Munich, Paris, Madrid, Cape Town, Sydney, Tokyo, Singapore, Mexico City, 2012, 603 p. ISBN 13: 978-0321749642, ISBN 10: 0321749642.
  11. OpenCL Optimization Guide. Available at: http://developer.amd.com/tools–and–sdks/opencl–zone/amd–accelerated–parallel–processing–app–sdk/opencl–optimization–guide/ (accessed 23.01.2015).
  12. Gaster B.R., Howes L., Kaeli D.R., Mistry P., Schaa D. Heterogeneous Computing with OpenCL, Morgan Kaufmann, Amsterdam, Boston, Heidelberg, London, New York, Oxford, Paris, San Diego, San Francisco, Singapore, Sydney, Tokyo, 2012, 277 p. ISBN 978-0-12-87766-6.
  13. Han T.D., Abdelrahman T.S. Reducing Branch Divergence in GPU Programs. In: GPGPU-4: Proceedings of the Fourth Workshop on General Purpose Processing on Graphics Processing Units, Newport Beach, California, USA, March 05, 2011, article no. 3. DOI: 10.1145/1964179.1964184.
  14. What is OpenMP? PARALLEL.RU. Available at: https://parallel.ru/tech/tech_dev/openmp.html (accessed 11.02.2015).
  15. OpenMP. Available at: http://www.openmp.org/ (accessed 11.02.2015).
  16. Chandra R., Dagum L., Kohr D., Maydan D., McDonald J., Melon R. Parallel Programming in OpenMP. Morgan Kaufmann Publishers, San Francisco, 2001, 231 p. ISBN 1-55860-671-8.
  17. OpenACC. Directives for Accelerators. Available at: http://www.openacc.org/ (accessed 21.04.2015).
  18. 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.
  19. Fedotov V.P., Spevak L.F., Nefedova O.A. Elastic-plastic deformation processes simulated by the modified boundary element method. Programmnye producty i sistemy, 2013, vol. 104, no. 4, pp. 253–257. DOI: 10.15827/0236-235X.104.253-257. (In Russian).
  20. 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 producty i sistemy, 2014, vol. 108, no. 4, pp. 178–182. DOI: 10.15827/0236-235X.108.178-182. (In Russian).
  21. 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, pp. 6918–6928. DOI: 10.1016/j.apm.2013.02.026.
  22. Spevak L.F., Nefedova O.A. Solving the nonlinear heat conduction equation by the dual reciprocity boundary element method. Mezhdunarodnyy zhurnal prikladnykh i fundamentalnykh issledovaniy, 2015, no. 12–1. Available at: http://www.applied–research.ru/ru/article/view?id=7813 (accessed: 05.12.2016). (In Russian).
  23. 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, pp. 1333–1343. DOI: 10.1016/j.apm.2015.06.038.
  24. Vazquez J.L., ed. The Porous Medium Equation: Mathematical Theory. Clarendon Press, Oxford, 2006, 648 p. ISBN 978-0-19-856903-9. DOI: 10.1093/acprof:oso/9780198569039.001.0001.
  25. Brebbia C.A., Telles J.F.C., Wrobel L.C. Boundary Element Techniques. Springer-Verlag, Berlin, Neidelberg, New-York, Tokyo, 1984, 466 p. ISBN: 978-3-642-48862-7 (Print), 978-3-642-48860-3 (Online). DOI: 10.1007/978-3-642-48860-3.
  26. Nardini D., Brebbia C.A. A New Approach to Free Vibration Analysis using Boundary Elements. Applied Mathematical Modelling, 1983, vol. 7, no. 3, pp. 157–162. DOI: 10.1016/0307-904X(83)90003-3.
  27. GSL-GNU Scientific Library. Available at: http://www.gnu.org/software/gsl/ (accessed 23.05.2016).
  28. Boost C++ Libraries. Available at: http://www.boost.org.


Article reference

Spevak L. F., Nefedova O. A. Parallelizing the Solution of the Nonlinear Heat Conduction Problem with the Application of the Opencl Library [Electronic resource] // 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://eng.dream-journal.org/issues/2016-6/2016-6_113.html
(accessed: 12/07/2022).  


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