Singularities in Differential Algebraic Boundary Value Problems Governing the Excitation Response of Beam Structures

Abstract

The objective of this paper2 is to identify and, where possible, resolve singularities that may arise in the discretization of spatiotemporal boundaryvalue problems governing the steadystate behavior of nonlinear beam structures. Of particular interest is the formulation of nondegenerate continuation problems of a geometricallynonlinear model of a slender beam, subject to a uniform harmonic excitation, which may be analyzed numerically in order to explore the parameterdependence of the excitation response. In the instances of degeneracy investigated here, the source is either found (i) directly in a differentialalgebraic system of equations obtained from a finiteelementbased spatial discretization of the governing partial differential boundaryvalue problem(s) together with constraints on the trial functions or (ii) in the further collocationbased discretization of the timeperiodic boundaryvalue problem. It is shown that several candidate spatial finiteelement discretizations of a mixed weak formulation of the governing boundaryvalue problem either result in (i) spatial group symmetries corresponding to equivariant vector fields and oneparameter families of periodic orbits along the group symmetry orbit or (ii) temporal group symmetries corresponding to ghost solutions and indeterminacy in a subset of the field variables. The paper demonstrates several methods for breaking the spatial equivariance, including projection onto a symmetryreduced state space or the introduction of an artificial continuation parameter. Similarly, the temporal indeterminacy is resolved by an asymmetric discretization of the governing differentialalgebraic equations. Finally, in the absence of theoretical bounds, computation is used to estimate convergence rates of the different discretization schemes, in the case of numerical calibration experiments performed on equilibrium and periodic responses for a linear beam, as well as for the full nonlinear models.