Title of the article PARALLELIZING OF COMPUTATIONS ON A GRAPHICS PROCESSING UNIT FOR ACCELERATING BOUNDARY ELEMENT CALCULATIONS IN MECHANICS
Authors

SHERBAKOV Sergei S., D. Sc. in Phys. and Math., Prof., Chief Researcher of the Laboratory of Wear and Fatigue Pamage Mechanics of the Republican Computer Center of Mechanical Engineering, Joint Institute of Mechanical Engineering of the NAS of Belarus, Minsk, Republic of Belarus; Professor of the Department of Theoretical and Applied Mechanics, Belarusian State University, Minsk, Republic of Belarus, This email address is being protected from spambots. You need JavaScript enabled to view it.">This email address is being protected from spambots. You need JavaScript enabled to view it.

POLESTCHUK Mikhail M., Junior Researcher of the Research Laboratory of Applied Mechanics, Belarusian State University, Minsk, Republic of Belarus, This email address is being protected from spambots. You need JavaScript enabled to view it.">This email address is being protected from spambots. You need JavaScript enabled to view it.

MARMYSH Dzianis E., Ph. D. in Phys. and Math., Assoc. Prof., Associate Professor of the Department of Theoretical and Applied Mechanics, Belarusian State University, Minsk, Republic of Belarus, This email address is being protected from spambots. You need JavaScript enabled to view it.">This email address is being protected from spambots. You need JavaScript enabled to view it.

In the section MECHANICS OF DEFORMED SOLIDS
Year 2024
Issue 1(66)
Pages 80–85
Type of article RAR
Index UDK 539.3:519.6+519.85
DOI https://doi.org/10.46864/1995-0470-2024-1-66-80-85
Abstract In solving problems of computer modeling using various methods, accuracy and computational efficiency questions always arise. This study explores the application of two modifications of the boundary element method to solve the problem of potential distribution within a closed two-dimensional domain with a uniform potential distribution on its boundary. The first modification involves using three nonlinear shape functions instead of one. The second modification applies the Galerkin method to the boundary element approach with three nonlinear shape functions. The essence of this modification lies in the fact that the system of equations is formulated in integral form over the entire boundary element, rather than at collocation points. In addition to this, the paper describes and investigates the advantages and disadvantages of the smoothing modification applied to these approaches. Since the influence matrix consists of independently computable elements, parallelization of calculations using NVIDIA CUDA technology has been proposed to enhance computational efficiency, significantly accelerating the calculation of interaction matrix. The choice of this technology is advantageous due to the prevalence of NVIDIA graphics accelerators in almost every personal computer or laptop, as well as it is easy to use. The study presents an approach to the application of this technology and demonstrates the results, showing the acceleration of parallelized calculations which show the dependence on the number of boundary elements. A comparison of the efficiency of the selected technology when applied to two methods, collocation and Galerkin, is also presented, indicating a significant speedup of up to 22 times by computing the influence matrix of the boundary elements.
Keywords boundary element method, nonlinear shape functions, potential distribution, NVIDIA CUDA, collocation method, Galerkin method, algorithm parallelization, numerical modeling, computational acceleration, interaction matrix
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Bibliography
  1. Crouch S.L., Starfield A.M. Boundary element methods in solid mechanics. London, Boston, Sydney, George Allen & Unwin, 1983.
  2. Banerjee P.K., Butterfield R. Boundary element methods in engineering science. London, Mcgraw-Hill Book Company (UK) Limited, 1981.
  3. Xu Y., Jackson R.L. Boundary element method (BEM) applied to the rough surface contact vs. BEM in computational mechanics. Friction, 2019, vol. 7, iss. 4, pp. 359–371. DOI: https://doi.org/10.1007/s40544-018-0229-3.
  4. Barhoumi B. An improved axisymmetric convected boundary element method formulation with uniform flow. Mechanics& industry, 2017, vol. 18, no. 3. DOI: https://doi.org/10.1051/meca/2016064.
  5. Sherbakov S.S., Polestchuk M.M. Uskorenie granichno-elementnykh raschetov s pomoshchyu graficheskogo akseleratora dlya elementov s nelineynymi funktsiyami formy [Acceleration of boundary-element computing using graphics accelerator for the elements with nonlinear form functions]. Mechanics of machines, mechanisms and materials, 2019, no. 4(49), pp. 89–94 (in Russ.).
  6. Zhang J., Han L., Zhong Y., Dong Y., Lin W. Boundary element analysis for elasticity problems using expanding element interpolation method. Engineering computations, 2020, vol. 37, no. 1, pp. 1–20. DOI: https://doi.org/10.1108/EC-11-2018-0506.
  7. Sutradhar A., Paulino G.H., Gray L.J. Symmetric Galerkin boundary element method. Berlin, Heidelberg, Springer-Verlag, 2008. 276 p. DOI: https://doi.org/10.1007/978-3-540-68772-6.
  8. Parreira P., Guiggiani M. On the implementation of the Galerkin approach in the boundary element method. Computers& structures, 1989, vol. 33, iss. 1, pp. 269–279. DOI: https://doi.org/10.1016/0045-7949(89)90150-8.
  9. Marmysh D.E., Sherbakov S.S. Granichno-elementnoe modelirovanie napryazhennogo sostoyaniya pri vdavlivanii shtampa v poluprostranstvo [Boundary-element modeling of stressed state under pressing-in of the indenter into the half-space]. Aktualnye voprosy mashinovedeniya, 2018, iss. 7, pp. 204–206 (in Russ.).
  10. Sanders J., Kandrot E. CUDA by example: an introduction to general-purpose GPU programming. Boston, Addison-Wesley Professional, 2010. 320 p.
  11. Sherbakov S.S., et al. Rasparallelivanie granichno-elementnykh raschetov s ispolzovaniem metoda Galerkina i nelineynykh funktsiy formy [Parallelization of boundary element calculations using the Galerkin method and nonlinear shape functions]. Vesnik of Yanka Kupala State University of Grodno. Series 6. Engineering science, 2021, vol. 11, no. 2, pp. 41–48 (in Russ.).
  12. Sherbakov S.S., Polestchuk M.M. Granichno-elementnoe modelirovanie s primeneniem metoda Galerkina, nelineynykh funktsiy formy i tekhnologii CUDA [Boundary element modeling using the Galerkin method, nonlinear shape functions and CUDA technology]. Materialy mezhdunarodnoy nauchnoy konferentsii “13 Belorusskaya matematicheskaya konferentsiya” [Proc. international scientific conference “13th Belarusian mathematical conference”]. Minsk, 2021, part 2, pp. 100–101 (in Russ.).
  13. Sherbakov S.S., Polestchuk M.M. Uskorenie granichno-elementnykh raschetov dlya zamknutoy oblasti s ispolzovaniem nelineynykh funktsiy formy i tekhnologii CUDA [Acceleration of boundary element calculations for closed domain using nonlinear form functions and CUDA technology]. Doklady BGUIR, 2021, vol. 19, no. 3, pp. 14–21. DOI: https://doi.org/10.35596/1729-7648-2021-19-14-21 (in Russ.).
  14. Rauber T., Rünger G. General purpose GPU programming. Parallel programming for multicore and cluster systems, 2013, pр. 387–415. DOI: https://doi.org/10.1007/978-3-642-37801-0_7.
  15. Storti D., Yurtoglu M. CUDA for engineers. An introduction to high-performance parallel computing. New York, Addison-Wesley, 2016. 328 p.