Numerical techniques for linear and nonlinear eigenvalue problems in the theory of elasticity

Abstract

Consider algorithms to solve eigenvalue problems for partial differential equations describing the bending of a von Karman elastic plate. Here we explore numerical techniques based on a variational principle, Newton's iterations and numerical continuation. The variational approach uses the Galerkin spectral method. First we study the linearized problem. Second, eigenfunctions of the nonlinear equations describing post-buckling behaviour of the von Karman plate are calculated. The plate is supposed to be totally clamped and compressed along its four sides. The basis functions in the variational procedure are trigonometric functions. They are chosen to match the boundary conditions. Effective computational techniques allow us to detect bifurcation points and to trace branches of solutions. Numerical examples demonstrate the efficiency of the methods. The proposed algorithms are applicable to similar problems involving elliptic differential equations.