FOURIER-SERIES FOR POLYGONAL PLATE-BENDING - A VERY LARGE PLATE ELEMENT

In this study, we developed an edge function approach using the Fourie r series for boundary value problems on polygonal domains. The method was then applied to classical plate bending problems. In subsequent wo rk, this method has been extended to the moderate rotation equations f or a shallow shell. Preliminary results indicate that the stability of a curved surface plays an important role in the morphogenesis of plan ts. For a polygonal plate with a convex domain, the Levy-type solution s for each edge serve as a set of fundamental functions. The set is co mplete and each member satisfies the equation exactly. The problem is solved by superimposing the solution functions and matching the Fourie r 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 ind ividual point sources. An extra harmonic is added so that corner bound ary conditions can be treated more efficiently. By this approach, a co nvex polygon is one element. Nonconvex domains, however, are divided i nto convex subdomains with appropriate continuity conditions at the in terfaces. 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 ex tended to the shallow shells.