Keywords: parameter identification, discrete wavelet transform, finite element analysis, back analysis, ill-posed problems, elastic constants, damage tensors.

Parameter identification plays an essential role in geotechnical engineering because we treat natural materials such as soils or rocks whose material properties are usually unknown. The material parameter identification procedures in geomechanics based on deterministic approaches are classified into inverse and direct approaches [1]. In the inverse approach, the system of equations governing the problem is rewritten in such a way that material parameters appear as unknowns and measured data as input data. On the other hand, the direct approach uses trial values of the unknown parameters as input data in the algorithm, until the discrepancy between measurements and numerical results is minimized. The proposed method combines the inverse approach and the direct approach. In the inverse part of this algorithm, the equilibrium equation is coupled together with observational boundary conditions, and the direct part of the algorithm improves the convergence of the Newton iteration process.

Most of the inverse problems generally require the solution of an ill-posed system of equations. Especially for a problem in which the number of unknown parameters exceeds the measured data, it is difficult to identify the unknown parameters.

The wavelet transform is a mathematical tool and has been widely used for image compression and signal processing. Recently, various finite element or boundary element techniques together with wavelet transforms have been studied for solving systems of linear equations. Doi et al. [2] presented a new inverse method using wavelet analysis in magnetic fields, which utilizes the data compression ability and the spectrum resolution ability of the waveforms.

In this paper, the above idea is applied to elastic materials [3] and damage mechanics problems. Two-dimensional discrete wavelet analysis is applied to the system matrix for identifying elastic materials and damage tensors of jointed rock masses. The accuracy of the results depends on the wavelet basis functions and the data compression ratio of the wavelet spectrum.

Layered vertical slope examples are calculated to investigate the validity and accuracy of the method. The numerical tests reveal that the parameter identification method using the discrete wavelet transform is applicable for not only determined systems where the number of unknown parameters m and the number of equations n are equal (n=m) and overdetermined (n>m) systems, but also underdetermined (n<m) systems.

1

G. Swoboda, Y. Ichikawa, Q. Dong, M. Zaki, "Back Analysis of Large Geotechnical Models", Int. J. Numer. Meth. Geomech., 23, 1455-1472, 1999. doi:10.1002/(SICI)1096-9853(199911)23:13<1455::AID-NAG33>3.3.CO;2-3

2

T. Doi, S. Hayano, Y. Saito, "Wavelet Solution of The Inverse Source Problems", IEEE Transactions on Magnetics, 33(2), 1935-1938, 1997. doi:10.1109/20.582671

3

T. Ohkami, J. Nagao, S. Koyama, "Identification of elastic materials using wavelet transform", Computers and Structures, 84, 1866-1873, 2006. doi:10.1016/j.compstruc.2006.08.030

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