Direct and discretized perturbations revisited: A new error source interpretation, with application to moving boundary problem

It is well-known that qualitative and/or quantitative differences will be possibly induced, when applying direct (i.e., attacking directly the partial differential equations) or discretized (i.e., using a low-order Galerkin truncated model) perturbation methods to nonlinear structures using multi-scale expansions, especially for spatially continuous structures dominated by a quadratic nonlinearity (structures with initial curvature, i.e., buckled beam, shallow arch, sagged cable, etc). In this paper, a new mechanism associated with structure's non-zero (non-homogeneous) boundary motion and cubic nonlinearity, rather than quadratic one, will be shown to cause also differences if applying the two versions of multiple time scale method to the same nonlinear system. Inspired by the two different nonlinear sources above, i.e., quadratic and cubic, a fundamental interpretation of differences/errors between the two perturbation formulations is proposed, with the above two kinds of nonlinearity being included. Explicitly, the full-spectrum forced solution at lower-order and the following high-order cross-interactions with the structural modes are captured by the direct perturbation, but not by the discretized perturbation, if a finite mode truncation is used. Also two different correction schemes are proposed for eliminating the errors above, i.e., full basis Galerkin discretization and a rational drift function using structural Green's function, respectively. •A new error source interpretation for direct and discretized perturbation formulations is proposed.•The incomplete low-order forced solutions and the induced cross-interactions cause erroneous perturbation results.•The perturbation error of structure’s moving boundary problem is well explained by employing the new interpretation.•Two correction schemes for perturbation analysis of structure's moving boundary problem are established.