Fourier series for polygonal plate bending: A very large plate element

Abstract

In this study, we developed an edge function approach using the Fourier series for boundary value problems on polygonal domains. The method was then applied to classical plate bending problems. In subsequent work, this method has been extended to the moderate rotation equations for a shallow shell. Preliminary results indicate that the stability of a curved surface plays an important role in the morphogenesis of plants. For a polygonal plate with a convex domain, the Lévy-type solutions for each edge serve as a set of fundamental functions. The set is complete and each member satisfies the equation exactly. The problem is solved by superimposing the solution functions and matching the Fourier harmonics of the prescribed boundary conditions. The process is much like the boundary element method (BEM), except that the unknowns are the amplitudes of Fourier harmonics, rather than the weightings of individual point sources. An extra harmonic is added so that corner boundary conditions can be treated more efficiently. By this approach, a convex polygon is one element. Nonconvex domains, however, are divided into convex subdomains with appropriate continuity conditions at the interfaces. In this method, similar to the boundary integral method, no mesh generation is involved; the number of elements and the degrees of freedom is significantly smaller than in the finite element or finite difference methods. The advantage over boundary elements is that the matrices are generally well-conditioned and the approach is readily extended to the shallow shells.