SPECTRAL ELEMENT METHODS FOR NONLINEAR SPATIOTEMPORAL DYNAMICS OF AN EULER-BERNOULLI BEAM

Spectral element methods are high order accur rate methods which have been successfully utilized for solving ordinary and partial differenti al equations. In this paper the space-time spectral element (STSE) met hod is employed to solve a simply supported modified Euler-Bernoulli n onlinear beam undergoing forced lateral vibrations. This system was ch osen for analysis due to the availability of a reference solution of t he form of a forced Duffing's equation. Two formulations were examined : i) a generalized Galerkin method with Hermitian polynomials as inter polants both in spatial and temporal discretization (HHSE), ii) a mixe d discontinuous Galerkin formulation with Hermitian cubic polynomials as interpolants for spatial discretization and Lagrangian spectral pol ynomials as interpolants for temporal discretization (HLSE). The first method revealed severe stability problems while the second method exh ibited unconditional stability and was selected for detailed analysis. The spatial h-convergence rate of the HLSE method is of order alpha = p(s) + 1 (where p(s) is the spatial polynomial order). Temporal p-con vergence of the HLSE method is exponential and the h-convergence rate based on the end points (the points corresponding to the final time of each element) is of order 2p(T) - 1 less than or equal to alpha less than or equal to 2p(T) + 1 (where pr is the temporal polynomial order) . Due to the high accuracy of the HLSE method, good results were achie ved for the cases considered using a relatively large spatial grid siz e (4 elements for first mode solutions) and a large integration time s tep (1/4 of the system period for first mode solutions, with p(T) = 3) . All the first mode solution features were detected including the ons et of the first period doubling bifurcation, the onset of chaos and th e return to periodic motion. Two examples of second mode excitation pr oduced homogeneous second mode and coupled first and second mode perio dic solutions. Consequently, the STSE method is shown to be an accurat e numerical method for simulation of nonlinear spatio-temporal dynamic al systems exhibiting chaotic response.