×

zbMATH — the first resource for mathematics

Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. (English) Zbl 1272.74623
Summary: This paper deals with numerical experiments related with the shape of the source contour in the application of the method of fundamental solutions to the elastic torsion of prismatic rods. The following five boundary-value problems (BVPs) connected with torsion are studied: L-section, [-section, +-section, \(\rfloor\!\!\lceil\)-section and I-section. For all five BVPs examined, the region of cross-section of rods is concave. Both the local and global errors are examined for two basic shapes of the source contour. In the first case, the source contour is a circle and in the second case the source contour is geometrically similar to the boundary contour of the region under consideration. Furthermore, the optimal radius of the source contour, in the case of the circle, or the optimal distance of the source contour from the boundary in the case it is geometrically similar, are studied. An influence of the method parameters (radius of the circle or distance between contours) on the condition linear system of equation is examined. In all examples examined the values of the local and global errors of the method are smaller when the source contour is geometrically similar to the boundary of the region under consideration in comparison to the source contour with a shape of a circle.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv comput math, 9, 69-95, (1998) · Zbl 0922.65074
[2] Fairweather, G.; Karageorghis, A.; Martin, P.A., The method of fundamental solutions for scattering and radiation problems, Eng anal bound elem, 27, 759-769, (2003) · Zbl 1060.76649
[3] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130
[4] Koopman, G.H.; Song, L.; Fahnline, J.B., A method for computing acoustic fields based on the principle of wave superposition, J acoust soc am, 86, 2433-2438, (1988)
[5] Cao, Y.; Schultz, W.; Beck, R., Three dimensional desingularized boundary integral methods for potential problems, Int J numer methods eng, 12, 785-803, (1991) · Zbl 0723.76069
[6] Burges, G.; Mahajerin, E., The fundamental collocation method applied to non-linear Poisson equation in two dimensions, Comput struct, 27, 763-767, (1987) · Zbl 0633.65115
[7] Katsurada, M.; Okamoto, H., A mathematical study of the charge simulation method. A mathematical study of the charge simulation method, J fac sci univ Tokyo, sect 1A, 35, 507-518, (1988) · Zbl 0662.65100
[8] Kupradze, V.D.; Aleksidze, M.A., Approximate method of solving certain boundary-value problems, Soobshch akad nauk gruz SSR, 30, 529-536, (1963), (in Russian)
[9] Kupradze, V.D.; Aleksidze, M.A., The method of functional equations for an approximate solution of certain boundary value problems, USSR comput math comput phys, 4, 683-715, (1964)
[10] Mathon, R.; Johnston, R.L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J numer anal, 14, 638-650, (1977) · Zbl 0368.65058
[11] Bogomolny, A., Fundamental solution method for elliptic boundary value problems, SIAM J numer anal, 22, 644-669, (1985) · Zbl 0579.65121
[12] Katsurada, M.; Okamoto, H., A mathematical study of the charge simulation method. A mathematical study of the charge simulation method, J fac sci univ Tokyo, sect 1A, 35, 507-518, (1988) · Zbl 0662.65100
[13] Katsurada, M., Asymptotic error analysis of the charge simulation method, J fac sci univ Tokyo, sect 1A, 37, 635-657, (1990) · Zbl 0723.65093
[14] Katsurada, M.; Okamoto, H., The collocation points of the fundamental solution method for the potential problem, Comput math appl, 31, 123-137, (1996) · Zbl 0852.65101
[15] Kitagawa, T., On the numerical stability of the method of fundamental solutions applied to the Dirichlet problem, Jpn J appl math, 35, 507-518, (1988)
[16] Kitagawa, T., Asymptotic stability of the fundamental solution method, J comp appl math, 38, 263-269, (1991) · Zbl 0752.65077
[17] Amano, K., A charge simulation method for the numerical conformal mapping of interior, exterior and doubly connected domains, J comp appl math, 53, 353-370, (1994) · Zbl 0818.30004
[18] Karageorghis, A.; Fairweather, G., The almansi of fundamental solutions for solving biharmonic problems, Int J numer methods eng, 26, 1668-1682, (1988) · Zbl 0639.65066
[19] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for numerical solution of the biharmonic equation, J comput phys, 69, 434-459, (1987) · Zbl 0618.65108
[20] Karageorghis, A.; Fairweather, G., The simple layer potential method of fundamental solutions for certain biharmonic equation, Int J numer methods fluids, 9, 1221-1234, (1989) · Zbl 0687.76028
[21] Cisilino, A.P.; Sensale, B., Application of a simulated annealing algorithm in the optimal placement of source points in the method of the fundamental solutions, Comput mech, 28, 129-136, (2002) · Zbl 1146.74347
[22] Nishimura, R.; Nishimori, K.; Ishihara, N., Determining the arrangement of fictious charges in charge simulation method using genetic algorithms, J electr, 49, 95-105, (2000)
[23] Nishimura, R.; Nishimori, K.; Ishihara, N., Automatic arrangement of fictitious charges and contour points in charge simulation method for polar coordinate system, J electr, 51-52, 618-624, (2001)
[24] Nishimura, R.; Nishihara, M.; Nishimori, K.; Ishihara, N., Automatic arrangement of fictitious charges and contour points in charge simulation method for two spherical electrodes, J electr, 57, 337-346, (2003)
[25] Saavedra, I.; Power, H., Adaptive refinement scheme for the least-squares approach of the method of fundamental solution for three-dimensional harmonic problems, Eng anal bound elem, 28, 1123-1133, (2004) · Zbl 1074.65132
[26] Mitic, P.; Rashed, Y.F., Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources, Eng anal bound elem, 28, 143-153, (2004) · Zbl 1057.65091
[27] Partridge, P.W.; Sensale, B., The method of fundamental solutions with dual reciprocity for diffusion and diffusion – convection using subdomains, Eng anal bound elem, 24, 633-641, (2000) · Zbl 1005.76064
[28] De Medeiros, G.C.; Partridge, P.W.; Brandao, J.O., The method of fundamental solutions with dual reciprocity for some problems in elasticity, Eng anal bound elem, 28, 453-461, (2004) · Zbl 1130.74482
[29] Kołodziej, J.A.; Kleiber, M., Boundary collocation method vs FEM for some harmonic 2-D problems, Comput struct, 33, 155-168, (1989) · Zbl 0708.73087
[30] Balakrishnan, K.; Ramachandran, K., The method of fundamental solutions for linear diffusion-reaction equations, Math comp mod, 31, 221-237, (2000) · Zbl 1042.35569
[31] Timoshenko, S.P.; Goodier, J.N., Theory of elasticity, (1951), McGraw-Hill New York, (Chapter 10) · Zbl 0045.26402
[32] Kolodziej JA. Zastosowanie metody kollokacji brzegowej w zagadnieniach mechaniki (in Polish, ang. Application of Boundary Collocation Method in Applied Mechanics Problems). Published by Poznan University of Technology; Poznan (Poland) 2001.
[33] Arutiumian, N.C.; Abramian, B.L., Kruczenije uprigich tiel (in Rusian, ang. torsion of elastic body), (1963), Gos. Izd. Fiz.-Mat. Lit Moskwa
[34] Chen, Y.-Z.; Chen, Y.-H., Solution of the torsion problem for bars with “L, +, [ T”-cross-section by a harmonic function continuation technique, Int J eng sci, 19, 791-804, (1981) · Zbl 0498.73052
[35] Chen, Y.-H., On a finite element model for solving Dirichlet’s problem of Laplace’s equation, Int J numer methods eng, 18, 687-700, (1982) · Zbl 0509.73074
[36] Collatz, L., The numerical treatment of differential equations, (1960), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.