The Earthquake Engineering Online Archive

On identification and prediction of dynamical systems governed by nonlinear hyperbolic integro-differential equations of Volterra type

Lin, I-Ho; Sackman, Jerome L.

UCB/SESM-1973/15, Dept. of Civil Engineering, University of California, Berkeley, 1973-12, 142 pages

A practical procedure is developed and tested for solving combined inverse and direct problems involving distributed systems and, in particular, characterizing the dynamic properties of a class of nonlinear viscoelastic materials whose constitutive equations are given by certain nonlinear Volterra integral equations. The method of lines is applied to approximate the governing nonlinear hyperbolic integro-differential equations by systems of nonlinear Volterra integro-differential equations, the Runge-Kutta-Pouzet scheme is implemented with automatic step size control for integrating such resulting systems, and Powell's method of minimization without calculating derivatives is employed to search for optimal parameters appearing in these equations. The direct, or prediction, problem is essentially resolved by means of the first two methods, and addition of the third completes the solution for the inverse, or identification, problem. Several one-dimensional wave propagation problems in linear and nonlinear materials are solved and the results are compared either with experiment or with those obtained by other methods. To test the procedure developed, many numerical experiments are performed using both uncorrupted and noisy data for nonlinear elastic, linear and nonlinear viscoelastic materials. Some wave propagation experiments are also conducted on a polyethylene specimen; by means of this procedure, an identification of its dynamic properties is carried out using data collected from these experiments and a prediction of its transient behavior is executed and compared with experimental measurements.

Available online: http://nisee.berkeley.edu/documents/SEMM/SEMM-73-15.pdf (14 MB)