Abstract
The efficient discretization of a multidimensional random field with high definition and large geometric size remains a significant challenge. Compared with the simulation of one-dimensional or two-dimensional random fields, the generation of three-dimensional (3-D) random fields using the traditional Karhunen–Loève (K–L) expansion method tends to require a relatively long computational time and a larger amount of physical memory. In the present study, a decomposed K–L expansion scheme is proposed, which is applicable when a separable autocorrelation function (ACF) is used. The separability in the ACF is a precondition for the implementation of the proposed method. The proposed method decomposes the discretization of a 3-D random field into that of separate one-dimensional random fields and computes the eigenpairs in each dimension respectively. The accuracy and efficiency of the proposed method are demonstrated, and the numerical solutions of the eigenpairs used for the random field discretization are validated by comparing them with the theoretical solutions. Comparisons with the traditional K–L expansion method showed that the proposed scheme significantly reduced the computational time and memory requirements. This makes it potentially useful for the discretization of a multidimensional random field with a large geometric size and high definition, which can be used in stochastic finite element analysis. The proposed random field discretization method was applied to analyze a 3-D saturated clay slope under undrained conditions. The results of the stochastic finite element analysis suggest that the 3-D model is rather advantageous for estimating slope stability when the spatial variability of soil properties is considered. The 3-D model can capture more spatial details in soil distribution and its effects on failure mechanisms and safety factors from a theoretical point of view. Therefore, the proposed decomposed K–L expansion method for 3-D random field discretization has strong application potential in geotechnical problems.
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Funding
This work was supported by the National Natural Science Foundation of China (52209150), China Postdoctoral Science Foundation (2022M723403), the Opening Fund of State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (LP2306), and the Fundamental Research Funds for the Central Universities (2022QN1026).
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Appendix
Appendix
The code used to replace the Kronecker product for the reduction of memory usage:
% The following calculation procedure is equivalent to ‘S = kron(Pz,kron(Px,Py))*X’ but it has a higher computation efficiency and takes a much lower memory usage.
% nx, ny, and nz are the grid-point numbers of the random field in the x, y, and z directions, respectively.
% Mx, My, and Mz are the truncated number of terms in the K–L expansion for the decomposed 1-D random fields in x, y, and z directions, respectively.
X = randn(Mx*My*Mz,1); % Mx*My*Mz–1 vector.
S = Py*reshape(X,My,Mz*Mx); % ny–Mz*Mx matrix.
S = reshape(S,ny,Mz,Mx); % ny–Mz–Mx array.
S = permute(S,[2,3,1]); % Mz–Mx–ny array.
S = Px*reshape(S,Mx,ny*Mz); % nx–ny*Mz matrix.
S = reshape(S,nx,ny,Mz); % nx–ny–Mz array.
S = permute(S,[2,3,1]); % ny–Mz–nx array.
S = Pz*reshape(S,Mz,nx*ny); % nz–nx*ny matrix.
S = reshape(S,nz,nx,ny); % nz–nx–ny array.
S = permute(S,[2,3,1]); % nx–ny–nz array.
S = reshape(S,nx*ny*nz,1); % nx*ny*nz–1 vector.
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Zhu, B., Hiraishi, T. A decomposed Karhunen–Loève expansion scheme for the discretization of multidimensional random fields in geotechnical variability analysis. Stoch Environ Res Risk Assess 38, 1215–1233 (2024). https://doi.org/10.1007/s00477-023-02625-8
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DOI: https://doi.org/10.1007/s00477-023-02625-8