Mathematical Modeling of Bending of a Circular Plate with the Use of S-Splines
The present paper is concerned with the application of newly developed high-order semi-local smoothing splines (or S-splines) in solving problems in elasticity. We will consider seventh-degree S-splines, which preserve the four continuous derivatives (C 4-smooth splines) and remain stable. The problem in question can be reduced to solving an inhomogeneous biharmonic equation by the Galerkin method, where as a system of basis functions we take the C 4-smooth fundamental S-splines. Such an approach is capable not only of delivering high accuracy of the resulting numerical solution under a fairly small number of basis functions, but may also easily deliver the sought-for loads. In finding the loads, as is known, one has to twice numerically differentiate the resulting bipotential, which is the solution of the biharmonic equation. This results in roundoff propagation.
KeywordsGalerkin Method Uniform Mesh Biharmonic Equation Partition Point Periodic Spline
Unable to display preview. Download preview PDF.
- 2.G. R. Kirchhoff, “Über das Gleichgewicht und die Bewegung einer elastischen Scheibe,” J. Reine Angew. Math., 40 (1850).Google Scholar
- 3.G. I. Marchuk and V. I. Agoshkov, Introduction into Projection-Grid Methods [in Russian], Nauka, Moscow (1987).Google Scholar
- 7.D. A. Silaev and J. G. Ingtem, “Semilocal smoothing splines of seventh degree,” Vestnik YuUrGU. Ser. Mat. Model. Progr., No. 6, 104–112 (2010).Google Scholar
- 8.D. A. Silaev and D. O. Korotaev, “S-spline on the disc,” in: Proc. Int. Conf. “Matematika. Komp’yuter. Obrazovanie”, Pushchino (2003), p. 157.Google Scholar
- 9.D. A. Silaev and D. O. Korotaev, “Solution of boundary-value problems using S-splines,” in Mathematics. Computer. Education, G. Yu. Reznichenko, ed., Regular and Chaotic Dynamics, Izhevsk (2006), pp. 85–104Google Scholar
- 10.D. A. Silaev and D. O. Korotaev, “Solving of boundary tasks by using S-spline,” Komp. Issled. Model., 1, No. 2, 161–172 (2009).Google Scholar
- 13.S. P. Timoshenko, History of the Strength of Materials, McGraw-Hill, New York (1953).Google Scholar