Modeling of three-dimensional elastic strain fields by point-source method

The work objective is to investigate the possibility and efficiency of three-dimensional numerical models of the elastic stress fields in the deformed solids. The field point-source method (PSM) designated as the method of fundamental solutions (MFS) in the foreign literature is used to develop these models. The PSM system generation for modeling fields of different physical nature is described. The concept of a point source of the elastic displacement field in the deformed solid is introduced. The research result is a developed PSM equations system that can be applied for solving three-dimensional problems in the elasticity theory, including the classical first and second boundary value problems in the elasticity theory (when either the voltage or bias is set on the boundary), as well as a mixed boundary problem (when on one part of the boundary, the displacement is set, and on the other - the voltage). The PSM properties are studied when solving standard problems, and the Dirichlet problem for a circular domain. The dependences of the numerical solution error on the problem parameters, in particular, on such as the charge number simulating the desired field, the remoteness of the charges from the solution domain boundaries, are obtained. The test problem of calculating the deformation field in the ball domain is solved. Upon the results obtained, the following conclusion is made. In the numerical solution of three-dimensional elasticity problems, a decreasing exponential dependence of the PSM error on the square root of the simulating charge number is observed. This property provides a numerical solution with a very low relative error that implies the PSM application perspectiveness in the numerical solution of the elasticity problems, including three-dimensional problems.

метод точечных источников, метод фундаментальных решений, задача теории упругости, задача Дирихле, field Point-Source method, method of fundamental solutions, elasticity problem, Dirichlet problem

1. Pobedrya, B.Е. Chislennye metody v teorii uprugosti i plastichnosti. [Numerical methods in the elasticity and plasticity theory.] 2nd ed. Moscow: Izd-vo MGU, 1995, 366 p. (in Russian).

2. Zenkevich, О., Morgan, K. Konechnye elementy i approksimatsiya. [Finite Elements and Approximation.] Moscow: Mir, 1986, 318 p. (in Russian).

3. Brebbia, C., Telles, J., Wrobel, L. Metody granichnykh elementov. [Boundary Element Techniques.] Moscow: Mir, 1987, 524 p. (in Russian).

4. Samarskiy, А.А. Teoriya raznostnykh skhem. [Difference scheme theory.] Moscow: Nauka, 1989, 616 p. (in Russian).

5. Aleksidze, М.А. Fundamental'nye funktsii v priblizhennykh resheniyakh granichnykh zadach. [Fundamental functions in approximate solutions of boundary value problems.] Moscow: Nauka, 1991, 352 p. (in Russian).

6. Bakhvalov, Yu.A., Knyazev, S.Yu., Shcherbakov, A.A. Matematicheskoe modelirovanie fizicheskikh poley metodom tochechnykh istochnikov. [Mathematical Modeling of Physical Fields by the Point-Source Method.] Bulletin of the Russian Academy of Sciences: Physics. 2008, vol. 72, no. 9, pp. 1259—1261 (in Russian).

7. Knyazev, S.Yu. Ustoychivost' i skhodimost' metoda tochechnykh istochnikov polya pri chislennom reshenii kraevykh zadach dlya uravneniya Laplasa. [Stability and convergence of the field point-source method in the numerical solution to boundary value problems for the Laplace equation.] Izvestia. Electromechanics. 2010, no. 3, pp. 3–12 (in Russian).

8. Fairweather, G., Karageorghis, A. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, 1998, vol. 9, pp. 69–95.

9. Golberg, M.A., Chen, C.S. The method of fundamental solutions for potential, Helmholtz and diffusion problems. Boundary Integral Methods — numerical and mathematical aspects. Southampton: Computational Mechanics Publication, 1998, pp. 103–176.

10. Bogomolny, A. Fundamental solutions method for elliptic boundary value problems. SIAM Journal on Numerical Analysis, 1985, vol. 22, no. 4, pp. 644–669.

11. Fairweather, G., Johnston, R.-L. The method of fundamental solutions for problems in potential theory. Treatment of Integral Equations by Numerical Methods. London: Academic Press, 1982, pp. 349–359.

12. Katsurada, M., Okamoto, H. The collocation points of the method of fundamental solutions for the potential problem. Computers & Mathematics with Applications, 1996, vol. 31, pp. 123–137.

13. Knyazev, S.Yu., Shcherbakova, E.E. Reshenie granichnykh zadach matematicheskoy fiziki metodom tochechnykh istochnikov polya. [Solution of boundary value problems of mathematical physics by the field Point-Source method.] Izvestia. Electromechanics. 2007, no. 3, pp. 11–15 (in Russian).

14. Knyazev, S.Yu., Shcherbakova, E.E. Reshenie zadach teplo- i massoperenosa s pomoshch'yu metoda tochechnykh istochnikov polya. [Solution to the problems of heat and mass transfer by the field point-source method.] Izvestia. Electromechanics. 2006, no. 4, pp. 43–47 (in Russian).

15. Knyazev, S.Yu., Pustovoyt, V.N., Shcherbakova, E.E. Modelirovanie poley uprugikh deformatsiy s primeneniem metoda tochechnykh istochnikov. [Modeling the elastic strain fields by point-source method.] Vestnik of DSTU, 2015, vol. 15, no. 1 (80), pp. 29–38 (in Russian).

16. Landau, L.D., Lifshits, E.M. Teoreticheskaya fizika. V 10 t. T. VII. Teoriya uprugosti. [Theoretical Physics. In 10 vol. Vol. VII. Elasticity theory.] Moscow: Nauka, 1987, 248 p. (in Russian).

17. Patterson, C., Sheikh, M.A. On the use of fundamental solutions in Trefftz method for potential and elasticity problems. Boundary Element Methods in Engineering. Berlin: Springer, 1982, pp. 973–980.

18. Redekop, D., Cheung, R.-S.-W. Fundamental solutions for the collocation method in three-dimensional elastostatics. Computers & Structures, 1987, vol. 26, pp. 703–707.

19. Yan Gu, Wen Chen, Xiaoqiao He. Improved singular boundary method for elasticity problems. Computers & Structures, 2014, vol. 135, pp. 7–82.

20. Marin, L., Karageorghis, A. The MFS–MPS for two-dimensional steady-state thermoelasticity problems. Engineering Analysis with Boundary Elements Journal Impact Factor & Information, 2013, vol. 37, iss. 7–8, pp. 1004–1020.

21. Christiansen, S. Condition number of matrices derived from two classes of integral equations. Mathematical Methods in the Applied Sciences, 1981, vol. 3, pp. 364–392.

22. Drombosky, T.W., Meyer, A.L., Ling, L. Applicability of the method of fundamental solutions. Engineering Analysis with Boundary Elements, 2009, vol. 33, pp. 637–643.

23. Smyrlis, Y.-S., Karageorghis, A. Some aspects of the method of fundamental solutions for certain harmonic problems. Journal of Scientific Computing, 2001, vol. 16 (3), pp. 341–371.