Symmetry and scaling properties of the von Kármán equations

Abstract

We study symmetries in the von Kármán equations with simply supported boundary conditions on rectangular domains. By embedding this fourth order plate problem into a space of periodic functions, we obtain hidden symmetries and scaling properties in its solution manifold. These properties are exploited for efficient numerical approximation of the solution branches at the bifurcation points, and are demonstrated with numerical examples.Mathematics Subject Classification (1991). 35B32, 73H05, 65N25, 65N06.

Application of the Lanczos algorithm for solving the linear systems that occur in continuation problems

“…In practice, one can implement the so-called selective reorthogonalization process, see e.g. Reference [11,Chapters 13,24] for details.…”

“…Figures 5 and 6 show the solution curve and its contour at ≈ 19:6555, respectively. Next, we used the restarted Lanczos-Galerkin method to solve the two linear systems that appeared in (24). The latter is more e cient than the former, since the average number of iterations required in the corrector process is almost reduced by half.…”

“…By doing this, a large amount of computational cost can be saved, see, e.g. References [24,18]. Probably the only price to pay is that the discretization matrix B corresponding to the partial di erential operator is quasi-symmetric [25, p. 106].…”