Data-driven approach to solve vertical drain under time-dependent loading


Currently, the vertical drain consolidation problem is solved by numerous analytical solutions, such as time-dependent solutions and linear or parabolic radial drainage in the smear zone, and no artificial intelligence (AI) approach has been applied. Thus, in this study, a new hybrid model based on deep neural networks (DNNs), particle swarm optimization (PSO), and genetic algorithms (GAs) is proposed to solve this problem. The DNN can effectively simulate any sophisticated equation, and the PSO and GA can optimize the selected DNN and improve the performance of the prediction model. In the present study, analytical solutions to vertical drains in the literature are incorporated into the DNN—PSO and DNN—GA prediction models with three different radial drainage patterns in the smear zone under time-dependent loading. The verification performed with analytical solutions and measurements from three full-scale embankment tests revealed promising applications of the proposed approach.

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c h :

coefficient of consolidation in the horizontal direction

c v :

coefficient of consolidation in the vertical direction

F c :

parameter to reflect the effect of the decay pattern

H :

soil thickness

k d :

coefficient of permeability of the drain

k h :

horizontal coefficient of permeability

k s :

horizontal coefficient of permeability of the smeared zone

k v :

vertical coefficient of permeability

m v :

coefficient of volume compressibility

q w :

discharge capacity

r d :

drain radius

r e :

equivalent radius of the influence zone

r s :

smeared zone radius

u :

excess pore water pressure

\(\bar{u}\) :

average excess pore water pressure


the surcharge load

ε v :

vertical strain

γ w :

unit weight of water


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Financial support provided by Ho Chi Minh City Open University is gratefully acknowledged.

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Correspondence to Thanh Cuong-Le.

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Nghia-Nguyen, T., Kikumoto, M., Khatir, S. et al. Data-driven approach to solve vertical drain under time-dependent loading. Front. Struct. Civ. Eng. 15, 696–711 (2021).

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  • vertical drain
  • artificial neural network
  • time-dependent loading
  • deep learning network
  • genetic algorithm
  • particle swarm optimization