Digital simulation of random processes and its applications

With the aid of the finite element method, the present paper deals with the problem of structural response variability resulting from the spatial variability of material properties of structures, when they are subjected to static loads of a deterministic nature. The spatial variabilities are modeled as two‐dimensional stochastic fields. The finite element discretization is performed in such a way that the size of each element is sufficiently small. Then, the present paper takes advantage of the Neumann expansion technique in deriving the finite element solution for the response variability within the framework of the Monte Carlo method. The Neumann expansion technique permits more detailed comparison between the perturbation and Monte Carlo solutions for accuracy, convergence, and computational efficiency. The result from such a Monte Carlo method is also compared with that based on the commonly used perturbation method. The comparison shows that the validity of the perturbation method is limited to the c...

Neumann Expansion for Stochastic Finite Element Analysis

Auto-regressive moving-average (ARMA) models of the same order for AR and MA components are used for the characterization and simulation of stationary Gaussian multivariate random processes with zero mean. The coefficient matrices of the ARMA models are determined so that the simulated process will have the prescribed correlation function matrix. To accomplish this, the two-stage least squares method is used. The ARMA representation thus established permits one, in principle, to generate sample functions of infinite length and with such a speed and computational mode that even real time generations of the sample functions can be easily achieved. The numerical example indicates that the sample functions generated by the method presented herein reproduce the prescribed correlation function matrix extremely well despite the fact that these sample functions are all very long. This is seen from the closeness between the analytically prescribed auto- and cross-correlation functions and the corresponding sample correlations computed from the generated sample functions.

ARMA Representation of Random Processes

Dealing with the issues of response variability for statically determinate structures, this study analyzes the response variability in two cases in which the spectral density function of the stochastic field takes limiting shapes. In these limiting cases, the spectral density, having a constant total area, concentrates sharply around the origin in one case and spreads thinly throughout in the other. Also, the present study derives the upper and lower bounds on the response variability. These results provide important physical as well as numerical insight into the response variability issue, whether we solve the problem by exact or numerical integration of equations of equilibrium or motion, or by other numerical methods. It is rather difficult to estimate experimentally the autocorrelation or spectral density function for the stochastic variation of material properties. In view of this, the upper bound results are particularly important, since the bounds derived here do not require knowledge of the autoco...

Structural Response Variability III

A Monte Carlo technique is presented which can effectively be used for nonlinear response analysis of a structure subjected to a random pressure field undergoing large deflections. The pressure field is idealized as a multidimensional Gaussian process with mean zero and homogeneous both in time and space. The response analysis is performed in the time domain by numerically simulating generalized forces rather than in the frequency domain. The solution satisfies the boundary conditions and the differential equation in a Galerkin sense. Two numerical examples involving large deflections of a string and a plate are worked out. The result indicates that the present method indeed provides a powerful tool in solving nonlinear structural response problems under random excitations.

Monte Carlo Solution of Nonlinear Vibrations

The vibration of a flexible elastic plate is investigated by a Monte Carlo technique. The response analysis is performed in time domain by numerically simulating the resulting generalized forces. The nonlinear plate deflection and mutual interaction between the plate motion and external and/or internal airflow is included. The fluid perturbations due to the structural motion are described by linear aerodynamic theory. The boundary-layer pressure field is idealized as a homogeneous multidimensional Gaussian random process with mean zero. The plate differential equations of motion and boundary conditions are satisfied in a Galerkin sense by developing a modal solution. Illustrative examples are presented for subsonic and supersonic flow regions using shallow cavity and two plate mode approximations.

C. M. Jan

Papers

Nonlinear Panel Response from a Turbulent Boundary Layer

Nonlinear panel response and noise transmission from a turbulent boundary layer by a Monte Carlo approach