Stochastic reduced order models for uncertain nonlinear dynamical systems

A general methodology is presented for the consideration of both data and model uncertainty in the determination of the response of geometrically nonlinear structural dynamic systems. The approach is rooted in the availability of reduced order models of these nonlinear systems with a deterministic basis extracted from a reference model (the mean model). Uncertainty, both from data and model, is introduced by randomizing the coefficients of the reduced order model in a manner that guarantees the physical appropriateness of every realization of the reduced order model, i.e. while maintaining the fundamental properties of symmetry and positive definiteness of every such reduced order model. This randomization is achieved not by postulating a specific joint statistical distribution of the reduced order model coefficients but rather by deriving this distribution through the principle of maximization of the entropy constrained to satisfy the necessary symmetry and positive definiteness properties. Several desirable features of this approach are that the uncertainty can be characterized by a single measure of dispersion, affects all coefficients of the reduced order model, and is computationally easily achieved. The reduced order modeling strategy and this stochastic modeling of its coefficients are presented in details and several applications to a beam undergoing large displacement are presented. These applications demonstrate the appropriateness and computational efficiency of the method to the broad class of uncertain geometrically nonlinear dynamic systems. INTRODUCTION The need to include system uncertainty in dynamic analyses has long been recognized in the context of some specific problems. For example, the response of turbomachinery/engine bladed disks has been known since the late 1960’s (e.g. [1]) to be highly sensitive to small blade-to-blade variations in their material/geometrical properties. This lack of robustness has thus motivated numerous stochastic analyses in which the uncertainty/variations in blade properties was introduced through the representation of certain blade characteristics as random variables. This stochastic modeling has however typically been ad-hoc, i.e. only some of the blade properties were considered as random, most notably natural frequencies, based on demonstrated/perceived sensitivity. With predictive capabilities becoming always faster and allowing always more complex models, the limitations associated with the uncertainty in the parameters of the systems (data uncertainty, e.g. in the material properties) and in the computational modeling of the physical system (model uncertainty, e.g. in the finite element representation of fasteners, lap joints, etc. and the approximation of the physical geometry) now appear clearly in many areas of structural dynamics. Accordingly, it has become quite important to dispose of general methodologies for the inclusion of uncertainty in dynamic analyses, as opposed to the ad-hoc approaches used in the past, and a series of recent investigations have focused on devising such general techniques that are consistent with state-of-the-art computational tools. An attractive approach of this type for data uncertainty is the stochastic finite element method (see in particular [2]) in which the random fields characterizing both the uncertain material properties and the response of the system are described by polynomial chaos expansions. Then, given a complete characterization of the uncertain material properties, a similarly complete representation of the stochastic response is obtained. Note however that this probabilistic approach relies on a given computational model and thus does not allow the consideration of model uncertainty. A probabilistic approach that does include both data and model uncertainty has recently been devised (Soize, [3-5]) and applied/validated (see [6] for a review) on a variety of dynamic problems involving linear structures with possible additional local nonlinearities. The inclusion of data and model uncertainty is accomplished in reduced order models of the structure through an appropriate stochastic representation of the elements of its mass, damping, and stiffness matrices. The variations of these random matrices around a baseline model (referred to as the mean model) is characterized by a single measure of dispersion, as opposed to a large number of parameters from statistical distributions. Accordingly, this probabilistic approach has been referred to as nonparametric and thus exhibits the following advantageous properties: i) includes both model and data uncertainty, ii) is characterized by only a mean reduced order model and a measure of dispersion, and, iii) is computationally expedient because it relies on reduced order models for the Monte Carlo simulations typically involved in the stochastic analysis of uncertain systems. These important properties motivate the extension of the nonparametric approach to dynamic systems with distributed, geometric nonlinearity, which is the focus of the present investigation. This extension will rely in particular on recent developments in the formulation of reduced order models of geometrically nonlinear systems (e.g. [7-10]) and will be accomplished in the general framework of linearly elastic geometrically nonlinear structures which encompasses as special cases beams and plates with the von Karman strain definition. GEOMETRIC NONLINEAR FORMULATION While many of the classical structural dynamic problems involving geometric nonlinearity relate to beam, plates, and shells in which the von Karman strain definition is used, it is of interest here to demonstrate the general applicability of the nonparametric stochastic modeling approach. To this end, an arbitrary linearly elastic (i.e. with a linear relation between the Green strain and second Piola-Kirchhoff stress tensors) structure undergoing large deformations will be considered in the sequel. The position vector of a point of the structure will be denoted by X in the reference configuration and as x in the deformed one so that the displacement vector is X x u   . The deformation gradient tensor F is then defined by its components ij F as j i ij j i ij X u X x F         (1) where ij  denotes the Kronecker symbol. Associated with the displacement field u are deformations which are characterized by the Green strain tensor E of components   ij kj ki ij F F E    2 1 . (2) Note in the above equation and in the ensuing ones that summation is implied on all repeated indices. The equation of motion of the structure is then given by (e.g. see [11])   i i jk ij k u b S F X   0 0 0       for 0   X (3) where S denotes the second Piola-Kirchhoff stress tensor, 0  is the density in the reference configuration, and 0 b is the vector of body forces, all of which are assumed to depend on the coordinates i X and be expressed in the reference configuration in which the structure occupies the domain 0  . The boundary, 0   , of the reference configuration domain 0  , is composed of two parts, t 0   on which the tractions 0 t are given and 0   on which the displacements are specified. Accordingly, the boundary conditions are 0 0 i k jk ij t n S F  for t X 0    (4) and 0  u for u X 0    . (5) Note in Eqs (3) and (4) that the vectors 0 b and 0 t correspond to the transport of the body forces and tractions applied on the deformed configuration, i.e. b and t, back to the reference configuration. This operation is accomplished through the relations b J b  0 and t dA da t        0 (6) where J is the Jacobian of the transformation   X x x  , i.e.   F J det  . Further, the area ratio dA da / can be expressed evaluated from [12] N F J n dA da T   (7) where N is the unit normal vector to 0   at the boundary point X and n is its counterpart on the deformed configuration. To complete the formulation of the elastodynamic problem, it remains to specify the constitutive behavior of the material. In this regard, it will be assumed here that the structure may exhibit a given nonzero steady temperature distribution   X T . Then, adopting a linear elastic model between the Green strain and second Piola-Kirchhoff stress tensors yields the linear relation   ) (th kl kl ijkl ij E E C S   (8) where ) (th E denotes the strain tensor arising from the potential thermal effects. This tensor can be expressed as ) ( ) ( th th C T E  (9) where ) (th C is the thermal expansion tensor. Finally, the fourth order elasticity tensor C satisfies the symmetry conditions klij ijlk jikl ijkl C C C C    (10) and the positive definiteness property 0  kl ijkl ij A C A (11) for any second order tensor A . REDUCED ORDER MODELING The previous section has provided the governing equations for the infinite dimensional problem of determining the stress and displacement fields everywhere in the structure considered. Following the discussion of the introduction, it is next desired to construct finite dimensional reduced order models of Eqs (1)-(9) that can be used for a nonparametric stochastic modeling of uncertainty. Before introducing the basis for the reduction, it is necessary to express the problem in its weak form. To this end, denote by   X v v  a vector function of X that is sufficiently differentiable and such that 0  v on 0   . Then, the weak formulation of the geometric nonlinear elastodynamic problem of Eqs (3)-(5) is to find the displacement field u such that                   t ds t v X d b v X d S F X v X d u v i i i i jk ij k i i i 0 0 0 0 0 0 0 0   (12) is satisfied for all i v satisfying the above conditions. A reduced order model of the nonlinear geometric problem can then be obtained by assuming the displacement field u in the form