Projection method for simulation of conductive transfer
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Published:
Jul 25, 2023
Keywords:
local projection method, conductive, transfer, Laplace, Poisson equations.
Main Article Content
Abstract
The article deals with the processes of conductive transfer: diffusion, filtration and heat. Such processes are quite well described by the Laplace or Poisson equations with different conditions on the boundaries (Dirichlet and Neumann). There are quite a lot of numerical methods for solving such equations: difference methods, methods of finite and boundary elements. In some cases, there are problems of reflecting the domain of a function when the domain has a complex shape or is multiply connected. The way out is to use distributed computing. The purpose of the authors' research is to develop a universal methodology for solving such problems that are invariant to the described processes and to the configuration of objects. A method is proposed for significantly reducing the estimated time through the use of distributed computing. In accordance with the peculiarities of the methodology, the entire area is divided into a finite number of local areas in which solutions are sought, provided that the solutions in the areas are joined. Approximation of the solution in the local area can be carried out by almost any function. The only limitation is the absence of breaks in the function and its derivatives. The shape of local areas can be of various sizes and configurations. Local areas can overlap each other. Boundary conditions can be implemented in local areas. An original method for iterative matching of solutions in adjacent local domains is proposed. To evaluate the effectiveness of the proposed methodology, a number of test problems were solved. The Poisson equation was solved for regions of different shapes. Good agreement between analytical and numerical solutions is obtained. A numerical study of the ship's hull rolling in an incompressible ideal fluid was also carried out. The problem of vertical oscillations of a rectangular contour in deep water was solved. The Laplace equation was solved.Various boundary conditions were used: a condition on the surface of an undisturbed liquid, non-reflecting conditions on vertical virtual boundaries, a condition on a horizontal virtual boundary simulating the condition on deep water. The consistency of the methodology was assessed by comparing the results of computational and full-scale experiments.
Article Details
How to Cite
Chelabchi, V., Tuzova, I., Panchenko, T., Starodub, V., Tuzov, O., & Chelabchi, V. (2023). Projection method for simulation of conductive transfer. Herald of the Odessa National Maritime University, (70), 143-164. https://doi.org/10.47049/2226-1893-2023-3-143-164
Section
Project and program management
References
1. Popov V.V. Calculation methods: a summary of lectures for students of the Faculty of Mechanics and Mathematics. – K.: Vydavnycho-polihrafichnyi tsentr «Kyivskyi universytet» 2012. – 303 p.
2. Klymenko A.V., Zorina V.M. Theoretical foundations of heat engineering. Thermal engineering experiment: Handbook, МЕІ, – 2001.
3. Veselovskyi V.B, Dreus A.Yu., Siasiev A.V. Mathematical modeling and calculation methods of heat-technological processes: Training manual. – D.: Vyd-vo Dnipropetr. un-tu, 2004. – 248 p.
4. Chu H.-P., Chen C.-L. Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem // Communications in Nonlinear Science and Numerical Simulation. ‒ 2008. ‒ Vol. 13. № 8. – P. 1605-1614.
5. Toghaw P. Kaneko H., Bey K.S., Lenbury Y. Numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201. № 1-2. – P. 414-430.
6. Lcrcher F., Gassner G., Munz C.-D. An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Computational Physics. ‒ 2008. ‒ Vol. 227. № 11. – P. 5649-5670.
7. Christou М., Sophocleous С., Christov С. Numerical investigation of the nonlinear heat diffusion equation with high nonlinearity on the boundary // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201. № 1-2. – P. 729-738.
8. Cattaneo P.M. Dalstra M., Melsen B. The Finite Element Method: a Tool to Study Orthodontic Tooth Movement // J Dent Res. – 2005. – Vol. 84(5). – P. 428-433.
9. Chapko R., Johansson R. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant // InverseProblems in Science and Engineering. – 2009. – 17. – P. 871-883.
10. Chapko R., Johansson R. On some iterative methods based on boundary integrals for elliptic Cauchy problems in semi-infinite domains // Electronic Journal of Boundary Elements. – 2009. – 7. – P.1-12.
11. Mochurad L.I. Harasym Y.S., Ostudin B.A. Maximal using of specifics of some boundary problems in potential theory after their numerical analysis // International Journal of Computing. – 2009. – Vol. 8. – № 2. – P. 149-156.
12. Guo Ben-Yu, Shen Jie. Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval // Numer. Math. 2000. Vol. 86. № 4. – P. 635-654.
13. Christou М., Sophocleous С., Christov С. Gassner Munz C.-D. Numerical investigation of the nonlinear heat diffusion equation with high nonlinearity on the boundary // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201. № 1-2. – P. 729-738.
14. Lcrcher F., Nuraliev G., An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Compu- tational Physics. ‒ 2008. ‒ Vol. 227. № 11. – P. 5649-5670.
15. Nazirov, S.A., Anorova S.A. Study of Numeric Convergence of the Method of R-functions in Problems of Constraint Torsion [Text] // American Journal of Computational and Applied Mathematics. – 2012. – Vol.2. № 4.– P. 189-196.
16. Ovcharenko V.A., Podliesnyi S.V., Zinchenko S.M. Fundamentals of the finite element method and its application in engineering calculations: Study guide. – Kramatorsk:DDMA, 2008. – 380 p.
17. Merkt R.V., Chelabchi V.V., Chelabchi V.M., Kukishev I.A. Computational experiment. Dynamics of systems // Visnyk Odeskoho natsionalnoho mors- koho universytetu: Zbirnyk naukovykh prats. – Odesa: ONMU, 2014. – № 1(40). – P. 214-227.
18. ChelabchyV.V. Adapting of methods of asolution of applied problems to the distributed calculations. // Visnyk Natsionalnoho tekhnichnoho universytetu «Kharkivskyi politekhnichnyi instytut»: Zbirnyk naukovykh prats, Tematych- nyi vypusk «Systemnyi analiz, upravlinnia ta informatsiini tekhnolohii», ‒ Kharkiv: NTU«KhPI», ‒ 2004, №1. – P. 15-23.
19. Horban V.O., Horban I.M., Masiuk S.V., Nikishov V.I. Application of the boundary element method for the calculation of ship waves // Prykladna hidromekhanika. ‒ 2011. ‒ Т. 13, № 4. ‒ P. 22-29.
20. Horban V.O., Masiuk S.V. Numerical modeling of the hydrodynamic interaction of bodies moving in a liquid // Prykladna hidromekhanika.– 2006.– 8 (80), No 3.– P. 27-49.
21. Novak S.I., Chelabchi V.V. Adaptation of numerical methods to distributed computing / Tezy dopovidei naukovo-praktychnoi konferentsii «Informatsiini upravliaiuchi systemy ta tekhnolohii». – Sumy.: Drukarskyi dim «Papirus». – 2012. – P. 161-163.
22. Vugts J.H. The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface // International Shipbuilding Progress. ‒ 1968. ‒ V.15.N 167. – P. 251-276.
23. Samarskyi A.A., Vabyshchevych P.N. Mathematical modeling and compu- tational experiment. M.: YMM RAN, 2000. – 409 p.
24. Slettery J.S. Theory of transfer of momentum, energy and mass in continuous media. M.: Enerhyia, 1978. – 448 p.
25. Samarskyi A.A., Tykhonov A.A. Equations of mathematical physics. M.: Yzd- vo MHU, 2004. – 798 p.
26. Chu H.-P., Chen C.L. Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem // Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, 2008. – P. 1605- 1614.
27. Kaneko H., Bey K., Lenbury SY.P., Toghaw P. Numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201, No. 1-2. – P. 414-430.
28. Lcrcher F., Gassner G., Munz C.-D. An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Computational Physics, vol. 227, no. 11, 2008. – P. 5649-5670.
29. Christou М., Sophocleous С., Christov С. Numerical investigation of the non- linear heat diffusion equation with high nonlinearity on the boundary // App- lied Mathematics and Computation, vol. 201, no. 1-2, 2008. – P. 729-738.
30. Cattaneo P.M., Dalstra M., Melsen B. The Finite Element Method: a Tool to Study Orthodontic Tooth Movement, vol. 84(5), 2005. – P. 428-433.
31. Chapko R., Johansson B.T. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant // Inverse Problems in Science and Engineering, 17, 2009. – P. 871-883.
32. Chapko R., Johansson B.T. On some iterative methods based on boundary integrals for elliptic Cauchy problems in semi-infinite domains // Electronic Journal of Boundary Elements, 7, 2009. – P.1-12.
33. Mochurad L. I., Harasym S. B., Ostudin Y.A. Maximal using of specifics of some boundary problems in potential theory after their numerical analysis // International Journal of Computing, vol. 8, no 2, 2009. – P. 149-156.
34. Guo Ben, Yu, Shen Jie. Laguerre-Galerkin method for nonlinear partial diffe- rential equations on a semi-infinite interval // Numer. Math, vol. 86, no. 4, 2000. – P. 635-654.
35. Christou М., Sophocleous С., Christov С. Numerical investigation of the non- linear heat diffusion equation with high nonlinearity on the boundary // App- lied Mathematics and Computation, vol. 201, no. 1-2, 2008. – P. 729-738.
36. Lcrcher F. Gassner G., Munz C.D. An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Computational Physics, vol. 227, no. 11, 2008. – P. 5649-5670.
37. Nazirov, S. A., Nuraliev F.M., Anorova S.A. Study of Numeric Convergence of the Method of R-functions in Problems of Constraint Torsion // American Journal of Computational and Applied Mathematics, vol. 2, no 4, 2012. – P. 189-196.
38. Marchuk H.Y., Ahoshkov V.Y. Introduction to projection-grid methods. M.: Nauka, 1981. – 416 p.
39. Kukyshev Y.A. Using the projection-grid method in distributed computing / И.А. Кукишев // Sbornyk nauchnыkh trudov Sword, vol. 2, T 6. Odessa: KUPRYENKO, 2013. – P. 24-28.
40. Chelabchy V.V. Adapting of methods of a solution of applied problems to the distributed calculations. // Visnyk Natsionalnoho tekhnichnoho universytetu «Kharkivskyi politekhnichnyi instytut»: Zbirnyk naukovykh prats, Tematych- nyi vypusk «Systemnyi analiz, upravlinnia ta informatsiini tekhnolohii», ‒ Kharkiv: NTU«KhPI», no 4, ‒ 2004. – P. 15-23.
41. Chelabchy V.V. Organization of distributed computing in problems of wave hydromechanics // Sb. nauchnykh trudov po materyalam nauchno-prakty- cheskoi konferentsyy «Sovremennye problemy y puty ykh reshenyia v nauke, transporte, proyzvodstve y obrazovanyy», T. 12. Odessa:Chernomore, 2005. – P.41-44.
42. Chelabchy V.V. Modeling the rolling of shallow draft vessels in deep water // Sb. nauchnykh trudov mezhdunarodnoi nauchno-praktycheskoi konferentsyy «Sovremennыe problemy y puty ykh reshenyia nauke, transporte, proyzvod- stve y obrazovanyy ‘2009», T. 22. Odessa: Chernomore, 2009. – P. 17-20.
43. Novak S.I., Chelabchy V.V. Adaptation of numerical methods to distributed computing // Tezy dopovidei naukovo-praktychnoi konferentsii «Informatsiini upravliaiuchi systemy ta tekhnolohii». Sumy.: Drukarskyi dim «Papirus», 2012. – P. 161-163.
2. Klymenko A.V., Zorina V.M. Theoretical foundations of heat engineering. Thermal engineering experiment: Handbook, МЕІ, – 2001.
3. Veselovskyi V.B, Dreus A.Yu., Siasiev A.V. Mathematical modeling and calculation methods of heat-technological processes: Training manual. – D.: Vyd-vo Dnipropetr. un-tu, 2004. – 248 p.
4. Chu H.-P., Chen C.-L. Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem // Communications in Nonlinear Science and Numerical Simulation. ‒ 2008. ‒ Vol. 13. № 8. – P. 1605-1614.
5. Toghaw P. Kaneko H., Bey K.S., Lenbury Y. Numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201. № 1-2. – P. 414-430.
6. Lcrcher F., Gassner G., Munz C.-D. An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Computational Physics. ‒ 2008. ‒ Vol. 227. № 11. – P. 5649-5670.
7. Christou М., Sophocleous С., Christov С. Numerical investigation of the nonlinear heat diffusion equation with high nonlinearity on the boundary // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201. № 1-2. – P. 729-738.
8. Cattaneo P.M. Dalstra M., Melsen B. The Finite Element Method: a Tool to Study Orthodontic Tooth Movement // J Dent Res. – 2005. – Vol. 84(5). – P. 428-433.
9. Chapko R., Johansson R. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant // InverseProblems in Science and Engineering. – 2009. – 17. – P. 871-883.
10. Chapko R., Johansson R. On some iterative methods based on boundary integrals for elliptic Cauchy problems in semi-infinite domains // Electronic Journal of Boundary Elements. – 2009. – 7. – P.1-12.
11. Mochurad L.I. Harasym Y.S., Ostudin B.A. Maximal using of specifics of some boundary problems in potential theory after their numerical analysis // International Journal of Computing. – 2009. – Vol. 8. – № 2. – P. 149-156.
12. Guo Ben-Yu, Shen Jie. Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval // Numer. Math. 2000. Vol. 86. № 4. – P. 635-654.
13. Christou М., Sophocleous С., Christov С. Gassner Munz C.-D. Numerical investigation of the nonlinear heat diffusion equation with high nonlinearity on the boundary // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201. № 1-2. – P. 729-738.
14. Lcrcher F., Nuraliev G., An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Compu- tational Physics. ‒ 2008. ‒ Vol. 227. № 11. – P. 5649-5670.
15. Nazirov, S.A., Anorova S.A. Study of Numeric Convergence of the Method of R-functions in Problems of Constraint Torsion [Text] // American Journal of Computational and Applied Mathematics. – 2012. – Vol.2. № 4.– P. 189-196.
16. Ovcharenko V.A., Podliesnyi S.V., Zinchenko S.M. Fundamentals of the finite element method and its application in engineering calculations: Study guide. – Kramatorsk:DDMA, 2008. – 380 p.
17. Merkt R.V., Chelabchi V.V., Chelabchi V.M., Kukishev I.A. Computational experiment. Dynamics of systems // Visnyk Odeskoho natsionalnoho mors- koho universytetu: Zbirnyk naukovykh prats. – Odesa: ONMU, 2014. – № 1(40). – P. 214-227.
18. ChelabchyV.V. Adapting of methods of asolution of applied problems to the distributed calculations. // Visnyk Natsionalnoho tekhnichnoho universytetu «Kharkivskyi politekhnichnyi instytut»: Zbirnyk naukovykh prats, Tematych- nyi vypusk «Systemnyi analiz, upravlinnia ta informatsiini tekhnolohii», ‒ Kharkiv: NTU«KhPI», ‒ 2004, №1. – P. 15-23.
19. Horban V.O., Horban I.M., Masiuk S.V., Nikishov V.I. Application of the boundary element method for the calculation of ship waves // Prykladna hidromekhanika. ‒ 2011. ‒ Т. 13, № 4. ‒ P. 22-29.
20. Horban V.O., Masiuk S.V. Numerical modeling of the hydrodynamic interaction of bodies moving in a liquid // Prykladna hidromekhanika.– 2006.– 8 (80), No 3.– P. 27-49.
21. Novak S.I., Chelabchi V.V. Adaptation of numerical methods to distributed computing / Tezy dopovidei naukovo-praktychnoi konferentsii «Informatsiini upravliaiuchi systemy ta tekhnolohii». – Sumy.: Drukarskyi dim «Papirus». – 2012. – P. 161-163.
22. Vugts J.H. The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface // International Shipbuilding Progress. ‒ 1968. ‒ V.15.N 167. – P. 251-276.
23. Samarskyi A.A., Vabyshchevych P.N. Mathematical modeling and compu- tational experiment. M.: YMM RAN, 2000. – 409 p.
24. Slettery J.S. Theory of transfer of momentum, energy and mass in continuous media. M.: Enerhyia, 1978. – 448 p.
25. Samarskyi A.A., Tykhonov A.A. Equations of mathematical physics. M.: Yzd- vo MHU, 2004. – 798 p.
26. Chu H.-P., Chen C.L. Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem // Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, 2008. – P. 1605- 1614.
27. Kaneko H., Bey K., Lenbury SY.P., Toghaw P. Numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates // Applied Mathematics and Computation. ‒ 2008. ‒ Vol. 201, No. 1-2. – P. 414-430.
28. Lcrcher F., Gassner G., Munz C.-D. An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Computational Physics, vol. 227, no. 11, 2008. – P. 5649-5670.
29. Christou М., Sophocleous С., Christov С. Numerical investigation of the non- linear heat diffusion equation with high nonlinearity on the boundary // App- lied Mathematics and Computation, vol. 201, no. 1-2, 2008. – P. 729-738.
30. Cattaneo P.M., Dalstra M., Melsen B. The Finite Element Method: a Tool to Study Orthodontic Tooth Movement, vol. 84(5), 2005. – P. 428-433.
31. Chapko R., Johansson B.T. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant // Inverse Problems in Science and Engineering, 17, 2009. – P. 871-883.
32. Chapko R., Johansson B.T. On some iterative methods based on boundary integrals for elliptic Cauchy problems in semi-infinite domains // Electronic Journal of Boundary Elements, 7, 2009. – P.1-12.
33. Mochurad L. I., Harasym S. B., Ostudin Y.A. Maximal using of specifics of some boundary problems in potential theory after their numerical analysis // International Journal of Computing, vol. 8, no 2, 2009. – P. 149-156.
34. Guo Ben, Yu, Shen Jie. Laguerre-Galerkin method for nonlinear partial diffe- rential equations on a semi-infinite interval // Numer. Math, vol. 86, no. 4, 2000. – P. 635-654.
35. Christou М., Sophocleous С., Christov С. Numerical investigation of the non- linear heat diffusion equation with high nonlinearity on the boundary // App- lied Mathematics and Computation, vol. 201, no. 1-2, 2008. – P. 729-738.
36. Lcrcher F. Gassner G., Munz C.D. An explicit discontinuous galerkin scheme with local time-stepping for general unsteady diffusion equations // Journal of Computational Physics, vol. 227, no. 11, 2008. – P. 5649-5670.
37. Nazirov, S. A., Nuraliev F.M., Anorova S.A. Study of Numeric Convergence of the Method of R-functions in Problems of Constraint Torsion // American Journal of Computational and Applied Mathematics, vol. 2, no 4, 2012. – P. 189-196.
38. Marchuk H.Y., Ahoshkov V.Y. Introduction to projection-grid methods. M.: Nauka, 1981. – 416 p.
39. Kukyshev Y.A. Using the projection-grid method in distributed computing / И.А. Кукишев // Sbornyk nauchnыkh trudov Sword, vol. 2, T 6. Odessa: KUPRYENKO, 2013. – P. 24-28.
40. Chelabchy V.V. Adapting of methods of a solution of applied problems to the distributed calculations. // Visnyk Natsionalnoho tekhnichnoho universytetu «Kharkivskyi politekhnichnyi instytut»: Zbirnyk naukovykh prats, Tematych- nyi vypusk «Systemnyi analiz, upravlinnia ta informatsiini tekhnolohii», ‒ Kharkiv: NTU«KhPI», no 4, ‒ 2004. – P. 15-23.
41. Chelabchy V.V. Organization of distributed computing in problems of wave hydromechanics // Sb. nauchnykh trudov po materyalam nauchno-prakty- cheskoi konferentsyy «Sovremennye problemy y puty ykh reshenyia v nauke, transporte, proyzvodstve y obrazovanyy», T. 12. Odessa:Chernomore, 2005. – P.41-44.
42. Chelabchy V.V. Modeling the rolling of shallow draft vessels in deep water // Sb. nauchnykh trudov mezhdunarodnoi nauchno-praktycheskoi konferentsyy «Sovremennыe problemy y puty ykh reshenyia nauke, transporte, proyzvod- stve y obrazovanyy ‘2009», T. 22. Odessa: Chernomore, 2009. – P. 17-20.
43. Novak S.I., Chelabchy V.V. Adaptation of numerical methods to distributed computing // Tezy dopovidei naukovo-praktychnoi konferentsii «Informatsiini upravliaiuchi systemy ta tekhnolohii». Sumy.: Drukarskyi dim «Papirus», 2012. – P. 161-163.