Asymptotic Finite Elements Introducing the Method of Integral Multiple Scales

Abstract

The method of integral multiple scales (MIMS) is introduced and applied to linear and nonlinear beam models. Based on the method of multiple scales, MIMS is applied to the system Lagrangian and directly results in a system solution. An analytical solution approach is applied to a linear beam-string model to produce a system of linear differential equations that can be solved to produce an asymptotic solution. The true power of MIMS is then demonstrated through a finite element approach by using a set of parametric shape functions based on beam strings. Where the analytic methodology is limited to continuous systems, the finite element approach is easily applied to discontinuous systems providing an analysis method useful with distributed piezoelectric laminates. Both static and dynamic results are discussed. The use of the asymptotic shape functions in the MIMS asymptotic finite element method results in extremely high precision and provides a methodology that could provide a more efficient analytical tool for the development of highly compliant discontinuous systems.