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Reduction of Numerical Model in Some Geotechnical Problems

Artur Góral

Artur Góral

Division of Geotechnics and Engineering Structures, Department of Concrete Structures, Lodz University of TechnologyŁódź,
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Góral, Artur
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Marek Lefik

Marek Lefik

Division of Geotechnics and Engineering Structures, Department of Concrete Structures Lodz, University of TechnologyŁódź,
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Lefik, Marek
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Marek Wojciechowski

Marek Wojciechowski

Division of Geotechnics and Engineering Structures, Department of Concrete Structures Lodz, University of TechnologyŁódź,
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Wojciechowski, Marek
  
07 oct 2023
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Categoría del artículo: Special Issue 19th KKMGiIG
Publicado en línea: 07 oct 2023
Páginas: 333 - 349
Recibido: 01 mar 2023
Aceptado: 18 ago 2023
DOI: https://doi.org/10.2478/sgem-2023-0016
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© 2023 Artur Góral et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1:

Scheme of the approximation of known learning output data (targets) by the transformation of the input signal on input layer by trained ANN. Segments between nodes are symbolic representations of weights (multipliers) that modify the nodal activities before attributing it to the next node.
Scheme of the approximation of known learning output data (targets) by the transformation of the input signal on input layer by trained ANN. Segments between nodes are symbolic representations of weights (multipliers) that modify the nodal activities before attributing it to the next node.

Figure 2:

Synthetic representation of the results of the model solution in observation points in the form of the ANN. Figure 2.a. represents the ANN approximation of the FE solution for the model of reference while in Figure 2.b. the same scheme applies for the reduced model.
Synthetic representation of the results of the model solution in observation points in the form of the ANN. Figure 2.a. represents the ANN approximation of the FE solution for the model of reference while in Figure 2.b. the same scheme applies for the reduced model.

Figure 3:

Scheme of numerical solution of inverse problem defined for the reduced model. In training, input to the “inverse ANN” are displacements in the observation points—the direct solutions of the reduced model while targets are the values of the parameters for which these displacements have been computed. In recall mode, the well-trained ANN responds with correct model parameters for the set of displacements obtained at the input.
Scheme of numerical solution of inverse problem defined for the reduced model. In training, input to the “inverse ANN” are displacements in the observation points—the direct solutions of the reduced model while targets are the values of the parameters for which these displacements have been computed. In recall mode, the well-trained ANN responds with correct model parameters for the set of displacements obtained at the input.

Figure 4:

The complex ANN_1@(ANN_2−1) acts as a formula that assigns the parameters of the reduced model to the realistic parametric descriptions of the problem that is used for its FE solution.
The complex ANN_1@(ANN_2−1) acts as a formula that assigns the parameters of the reduced model to the realistic parametric descriptions of the problem that is used for its FE solution.

Figure 5:

Diagram of the considered layered pavement models. From top: model A; model B, C.
Diagram of the considered layered pavement models. From top: model A; model B, C.

Figure 6:

Examples of finite element meshes of the models.
Examples of finite element meshes of the models.

Figure 7:

Deflections of the pavement for the selected sets of stiffnesses. From left: model A, model B. Deflections between (from 2.4 to 4.0) [m] are not recorded.
Deflections of the pavement for the selected sets of stiffnesses. From left: model A, model B. Deflections between (from 2.4 to 4.0) [m] are not recorded.

Figure 8:

Direct Nk network learning results for two of all nine reading point of pavement deflection: 0 in title—1st reading point, 7 in title—8th reading point, target—reference deflection values, opt—deflection values identified by the trained ANN. On the horizontal axis—number of patterns. The target and output coincide (the blue line is not visible!)
Direct Nk network learning results for two of all nine reading point of pavement deflection: 0 in title—1st reading point, 7 in title—8th reading point, target—reference deflection values, opt—deflection values identified by the trained ANN. On the horizontal axis—number of patterns. The target and output coincide (the blue line is not visible!)

Figure 9:

Normalized response of the network versus the pattern deflection values: u_0—results for 1st reading point, u_1—results for 2nd reading point, u_2—results for 3rd reading point. The points on the graph also contain test data.
Normalized response of the network versus the pattern deflection values: u_0—results for 1st reading point, u_1—results for 2nd reading point, u_2—results for 3rd reading point. The points on the graph also contain test data.

Figure 10:

Scheme of the task of identifying the mechanical parameters of the surface and soil, description of symbols in the text.
Scheme of the task of identifying the mechanical parameters of the surface and soil, description of symbols in the text.

Figure 11:

Trial deflections computed in the frame of the reduced model.
Trial deflections computed in the frame of the reduced model.

Figure 12:

The results of learning the “inverse” network ANN_2−1. Targets—are values of mechanical properties to learn, opt—values of mechanical properties identified by the trained net.
The results of learning the “inverse” network ANN_2−1. Targets—are values of mechanical properties to learn, opt—values of mechanical properties identified by the trained net.

Figure 13:

Normalized response of the “inverse” network versus expected output. The points on the graph also contain test data.
Normalized response of the “inverse” network versus expected output. The points on the graph also contain test data.

Figure 14:

Reference (target) and ANN_3 identified (net_approx.) deflections of the pavement for three random test cases.
Reference (target) and ANN_3 identified (net_approx.) deflections of the pavement for three random test cases.

Figure 15:

Idealization of the interaction of the pavement and soil, the positive direction of the axis and load as well as the assumed material parameters are marked. A detailed description of this scheme—in the text of the article.
Idealization of the interaction of the pavement and soil, the positive direction of the axis and load as well as the assumed material parameters are marked. A detailed description of this scheme—in the text of the article.

Figure 16:

Comparison of the results of the network calculations with the known values of the material parameters of the Pasternak model: in Fig. 16.a, it is the Young's modulus of the stiffness of the E beam; in Figure 16.b it is the Winkler stiffness of the ground kW; in Figure 16.c it is Pasternak's constant G. These are the result of calculations in the reminder mode for a trained network with the structure of ANN_953, for 100 datasets, nine deflections each.
Comparison of the results of the network calculations with the known values of the material parameters of the Pasternak model: in Fig. 16.a, it is the Young's modulus of the stiffness of the E beam; in Figure 16.b it is the Winkler stiffness of the ground kW; in Figure 16.c it is Pasternak's constant G. These are the result of calculations in the reminder mode for a trained network with the structure of ANN_953, for 100 datasets, nine deflections each.

Summary of data for the applied model of the layered pavement_

Model A Model B Model C
Layer hj [cm] vj [-] Ej [MPa] hj [cm] vj [-] Ej [MPa] hj [cm] vj [-] Ej [MPa]
W0 15÷25 0,2 32000 15÷25 0,2 32000 15÷25 0,2 32000
W1 300 0,25 50÷200 100÷200 0,25 50÷200 20 0,17 9100÷13850
W2 - - - 300 0,25 50÷200 380 0,25 50÷200
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Geociencias, Geociencias, otros, Ciencia de los materiales, Compuestos, Materiales porosos, Física, Mecánica y dinámica de fluidos
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