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

Summary

This paper presents an identification method for material parameters using observational boundary conditions and wavelet analysis. The outline of the method is described for linear elastic problems, and it may be extended for visco-elastic, damage mechanics and elasto-plastic problems. The boundary of a material body $\Omega$ is divided into the displacement part $\partial \Omega_{u}$ and the traction part $\partial \Omega_{t}$ , so that $\partial \Omega_{u} \cup \partial \Omega_{t} =\partial \Omega$ . Then, corresponding to the applied load $\hat{ \mbox{\boldmath $t$}}$ on $\partial \Omega_{t}$ the displacements $\bar{ \mbox{\boldmath $u$}}$ are measured on this same part of the boundary and the displacement control condition can be obtained, and/or for the applied displacements $\hat{ \mbox{\boldmath $u$}}$ on $\partial \Omega_{u}$ the tractions $\bar{ \mbox{\boldmath $t$}}$ are measured on this same part of the boundary and the traction control boundary can be obtained.

The finite element method (FEM) for linear elastic problems in matrix form gives an algebraic equation in which the stiffness matrix involves the unknown material parameters . The parameters to be identified are implicitly included in the FEM governing equation combined with the observational boundary conditions.

Applying the Newton's iteration scheme for the FEM governing equation and the observational boundary conditions, a system equation for identifying the unknown parameters is derived [1,2]. If an inverse problem is overdetermined system, that is, there are more equations than unknowns ( the system matrix is then represented by a rectangular $n \times m$ matrix, where ), a non-linear least squares method may be introduced. However, if a problem is underdetermined where , it is difficult to determine the parameters .

Most of the inverse problems generally reduced into solving the ill-posed system of equations. Especially for a problem in which the number of unknown parameters exceed the measured data, it is difficult to identify the unknown parameters.

Doi et al. presented a new inverse method using the wavelet analysis in magnetic field [3,4], which utilizes the data compress ability and the spectrum resolution ability of the wave forms.

In this paper, we apply the two-dimensional discrete wavelet analysis to the system matrix of the basic equation for identifying parameter and find an approximate inverse matrix of the system matrix from the wavelet spectrum.

Applying the two dimensional discrete wavelet transform to the system equation, the system matrix is transformed into the wavelet spectrum $^{\prime}$ . It is known that the wavelet spectrum has large absolute values around the mother wavelet. Then, we extract a square matrix which is composed of the dominant elements from the entire wavelet spectrum $^{\prime}$ and calculate the inverse matrix $^{-1}$ . Combining $^{-1}$ with a zero rectangular matrix yields the approximate inverse matrix $^{\prime -1}_{Appro}$ in the wavelet spectrum space. Thus, an approximate inverse matrix $^{-1}_{Appro}$ of the system matrix is obtained by applying the inverse wavelet transform to the matrix $^{\prime -1}_{Appro}$ . Finally, the unknown vector can be obtained and a new vector is determined by means of the one-dimensional optimization method along the vector .

The validity of this method is numerically examined for multi-layered vertical slope problems in which the influence of the wavelet basis function and the data compressibility ratio on the accuracy is investigated.

From numerical examples, it is shown that accurate results will be obtained, if an order of the wavelet basis is set to a magnitude reduced to about one-sixth or one-seventh of the size of the wavelet transform matrix and a size of the wavelet spectrum is made into the larger dimension of rows and columns of the system matrix . As a conclusion of numerical calculations, the proposed method is shown to be effective for identifying the elastic constants even for the case in which the number of unknown parameters exceed the measured data.

References

1: G. Swoboda, Y. Ichikawa, Q. Dong and 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>3.0.CO;2-C
2: Y. Ichikawa and T. Ohkami, "A Parameter Identification Procedure as A Dual Boundary Control Problem for Linear Elastic Materials", Soils and Foundations, 32(2), 35-44, 1992.
3: T.Doi, S.Hayano and Y.Saito, "Wavelet Solution of The Inverse Source Problems", IEEE Transactions on Magnetics, 33(2), 1935-1938, 1997. doi:10.1109/20.582671
4: T.Doi, S.Hayano and Y.Saito, "Wavelet Solution of The Inverse Parameter Problems", IEEE Transactions on Magnetics, 33(2), 1962-1965, 1997. doi:10.1109/20.582678
5: T. Ohkami and G. Swoboda, "Parameter Identification of Viscoelastic Materials", Computers and Geotechnics, 24(4), 279-295, 1999. doi:10.1016/S0266-352X(99)00011-7
6: P. Dziadziuszko, Y. Ichikawa and Z. Sikora, "Inverse Analysis Procedure for Identifying Hardening Function in Elasto-plastic Problem by Two-stage Finite Element Scheme", Inverse Problems in Engineering, 8, 391-411, 2000. doi:10.1080/174159700088027737

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 77 PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping Paper 116 Parameter Identification Method using Wavelet Transform T. Ohkami, J. Nagao and S. Koyama Department of Architecture and Civil Engineering, Shinshu University, Nagano, Japan doi:10.4203/ccp.77.116 purchase the full-text of this paper Full Bibliographic Reference for this paper T. Ohkami, J. Nagao, S. Koyama, "Parameter Identification Method using Wavelet Transform", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 116, 2003. doi:10.4203/ccp.77.116 Keywords: parameter identification, wavelet transform, elastic constants, finite element analysis, back analysis, ill-posed problems. Summary This paper presents an identification method for material parameters using observational boundary conditions and wavelet analysis. The outline of the method is described for linear elastic problems, and it may be extended for visco-elastic, damage mechanics and elasto-plastic problems. The boundary of a material body $\Omega$ is divided into the displacement part $\partial \Omega_{u}$ and the traction part $\partial \Omega_{t}$ , so that $\partial \Omega_{u} \cup \partial \Omega_{t} =\partial \Omega$ . Then, corresponding to the applied load $\hat{ \mbox{\boldmath $t$}}$ on $\partial \Omega_{t}$ the displacements $\bar{ \mbox{\boldmath $u$}}$ are measured on this same part of the boundary and the displacement control condition can be obtained, and/or for the applied displacements $\hat{ \mbox{\boldmath $u$}}$ on $\partial \Omega_{u}$ the tractions $\bar{ \mbox{\boldmath $t$}}$ are measured on this same part of the boundary and the traction control boundary can be obtained. The finite element method (FEM) for linear elastic problems in matrix form gives an algebraic equation in which the stiffness matrix involves the unknown material parameters . The parameters to be identified are implicitly included in the FEM governing equation combined with the observational boundary conditions. Applying the Newton's iteration scheme for the FEM governing equation and the observational boundary conditions, a system equation for identifying the unknown parameters is derived [1,2]. If an inverse problem is overdetermined system, that is, there are more equations than unknowns ( the system matrix is then represented by a rectangular $n \times m$ matrix, where ), a non-linear least squares method may be introduced. However, if a problem is underdetermined where , it is difficult to determine the parameters . Most of the inverse problems generally reduced into solving the ill-posed system of equations. Especially for a problem in which the number of unknown parameters exceed the measured data, it is difficult to identify the unknown parameters. Doi et al. presented a new inverse method using the wavelet analysis in magnetic field [3,4], which utilizes the data compress ability and the spectrum resolution ability of the wave forms. In this paper, we apply the two-dimensional discrete wavelet analysis to the system matrix of the basic equation for identifying parameter and find an approximate inverse matrix of the system matrix from the wavelet spectrum. Applying the two dimensional discrete wavelet transform to the system equation, the system matrix is transformed into the wavelet spectrum $^{\prime}$ . It is known that the wavelet spectrum has large absolute values around the mother wavelet. Then, we extract a square matrix which is composed of the dominant elements from the entire wavelet spectrum $^{\prime}$ and calculate the inverse matrix $^{-1}$ . Combining $^{-1}$ with a zero rectangular matrix yields the approximate inverse matrix $^{\prime -1}_{Appro}$ in the wavelet spectrum space. Thus, an approximate inverse matrix $^{-1}_{Appro}$ of the system matrix is obtained by applying the inverse wavelet transform to the matrix $^{\prime -1}_{Appro}$ . Finally, the unknown vector can be obtained and a new vector is determined by means of the one-dimensional optimization method along the vector . The validity of this method is numerically examined for multi-layered vertical slope problems in which the influence of the wavelet basis function and the data compressibility ratio on the accuracy is investigated. From numerical examples, it is shown that accurate results will be obtained, if an order of the wavelet basis is set to a magnitude reduced to about one-sixth or one-seventh of the size of the wavelet transform matrix and a size of the wavelet spectrum is made into the larger dimension of rows and columns of the system matrix . As a conclusion of numerical calculations, the proposed method is shown to be effective for identifying the elastic constants even for the case in which the number of unknown parameters exceed the measured data. References 1 G. Swoboda, Y. Ichikawa, Q. Dong and 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>3.0.CO;2-C 2 Y. Ichikawa and T. Ohkami, "A Parameter Identification Procedure as A Dual Boundary Control Problem for Linear Elastic Materials", Soils and Foundations, 32(2), 35-44, 1992. 3 T.Doi, S.Hayano and Y.Saito, "Wavelet Solution of The Inverse Source Problems", IEEE Transactions on Magnetics, 33(2), 1935-1938, 1997. doi:10.1109/20.582671 4 T.Doi, S.Hayano and Y.Saito, "Wavelet Solution of The Inverse Parameter Problems", IEEE Transactions on Magnetics, 33(2), 1962-1965, 1997. doi:10.1109/20.582678 5 T. Ohkami and G. Swoboda, "Parameter Identification of Viscoelastic Materials", Computers and Geotechnics, 24(4), 279-295, 1999. doi:10.1016/S0266-352X(99)00011-7 6 P. Dziadziuszko, Y. Ichikawa and Z. Sikora, "Inverse Analysis Procedure for Identifying Hardening Function in Elasto-plastic Problem by Two-stage Finite Element Scheme", Inverse Problems in Engineering, 8, 391-411, 2000. doi:10.1080/174159700088027737 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £123 +P&P)
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