Identifying parameters of advanced soil models using an enhanced transitional Markov chain Monte Carlo method

Abstract

Parameter identification using Bayesian approach with Markov Chain Monte Carlo (MCMC) has been verified only for certain conventional simple constitutive models up to now. This paper presents an enhanced version of the differential evolution transitional MCMC (DE-TMCMC) method and a competitive Bayesian parameter identification approach for applying to advanced soil models. To realize the intended computational savings, a parallel computing implementation of DE-TMCMC is achieved using the single program/multiple data technique in MATLAB. To verify its robustness and effectiveness, synthetic numerical tests with/without noise and real laboratory tests are used for identifying the parameters of a critical state-based sand model based on multiple independent calculations. The original TMCMC is also used for comparison to highlight that DE-TMCMC is highly robust and effective in identifying the parameters of advanced sand models. Finally, the proposed parameter identification using DE-TMCMC is applied to identify parameters of an elasto-viscoplastic model from two in situ pressuremeter tests. All results demonstrate the excellent ability of the enhanced Bayesian parameter identification approach on identifying parameters of advanced soil models from both laboratory and in situ tests.

Appendix—introduction of SIMSAND

The selected critical state-based sand model was SIMSAND, which has seen widely used in various studies [25, 27, 28, 65]. Accordingly, only basic principles were introduced herein for the ease of understanding. Consistent with the elasto-plastic theory, the total strain rate is composed of the elastic and plastic strain rates:

$$\delta \varepsilon_{ij} = \delta \varepsilon_{ij}^{el} + \delta \varepsilon_{ij}^{pl}$$

(16)

where $\delta \varepsilon_{ij}$ denotes the (i, j) the total strain rate tensor and the superscripts el and pl represent the elastic and plastic components, respectively.

The nonlinear elastic behaviour is assumed to be isotropic with the Young’s modulus E:

$$\delta \varepsilon_{ij}^{el} = \frac{1 + \upsilon }{E}\delta \sigma^{\prime}_{ij} - \frac{\upsilon }{E}\delta \sigma^{\prime}_{kk} \delta_{ij}$$

(17)

where υ is Poisson’s ratio, $\delta \sigma^{\prime}_{ij}$ is the effective stress rate tensor, and $\delta_{ij}$ is the Kronecker delta. E is calculated by using the isotropic elastic bulk modulus K by E = 3 K(1 − 2υ), and for sand, is defined as follows:

$$K = K_{0} p_{\text{at}} \frac{{\left( {2.97 - e} \right)^{2} }}{{\left( {1 + e} \right)}}\left( {\frac{{p^{\prime}}}{{p_{\text{at}} }}} \right)^{n}$$

(18)

where K₀ and n are elastic parameters, e is void ratio, p′ is the mean effective stress, and p_at is the atmospheric pressure (p_at = 101.325 kPa).

The yield surface for shear sliding can be expressed as follows:

$$f = \frac{q}{{p^{\prime}}} - \frac{{M_{p} \varepsilon_{d}^{p} }}{{k_{p} + \varepsilon_{d}^{p} }}{ = 0 }$$

(19)

where q is the deviatoric stress, k_p is related to the plastic shear modulus, M_p is stress ratio corresponding to the peak strength calculated by the peak friction angle ϕ_p (M_p = 6sin(ϕ_p)/(3 − sin(ϕ_p)) in compression), and $\varepsilon_{d}^{p}$ is the deviatoric plastic strain.

The gradient of the plastic potential surface for stress-dilatancy $g$ can be expressed as follows:

$$\frac{\partial g}{{\partial \sigma_{ij} }} = \frac{\partial g}{{\partial p^{\prime}}}\frac{{\partial p^{\prime}}}{{\partial \sigma_{ij} }} + \frac{\partial g}{\partial q}\frac{\partial q}{{\partial \sigma_{ij} }}{\text{with }}\frac{\partial g}{{\partial p^{\prime}}} = A_{d} \left( {M_{pt} - \frac{q}{{p^{\prime}}}} \right); \, \frac{\partial g}{\partial q} = 1$$

(20)

where A_d is the stress-dilatancy parameter and M_pt can be calculated from the phase transformation friction angle ϕ_pt (M_pt = 6sin(ϕ_pt)/(3-sin(ϕ_pt)) in compression). The double index ij is simplified as $1{\hat{ = }}11, \, 2{\hat{ = }}22, \, 3{\hat{ = }}33, \, 4{\hat{ = }}12, \, 5{\hat{ = }}23, \, 6{\hat{ = }}31$.

A nonlinear critical state line (CSL) formulation to guarantee the positiveness of the critical void ratio was well suited to sand modelling

$$e_{\text{c}} = e_{\text{ref}} \exp \left[ { - \lambda \left( {\frac{{p^{\prime}}}{{p_{at} }}} \right)^{\xi } } \right]$$

(21)

where e_c is the critical void ratio, e_ref is the initial critical void ratio at p′ = 0, and λ and ξ are the parameters controlling the shape of CSL in the e-logp′ plane.

Soil density and the interlocking grains effects are introduced through the expression of the friction angle as follows:

$$\tan \phi_{p} = \left( {\frac{{e_{c} }}{e}} \right)^{{n_{p} }} \tan \phi_{u} ;\quad \tan \phi_{pt} = \left( {\frac{{e_{c} }}{e}} \right)^{{ - n_{d} }} { \tan }\phi_{\mu }$$

(22)

where the parameters n_p and n_d are material constants and ϕ_μ is friction angle at critical state. The Lode angle-dependent strength and stress-dilatancy are introduced as described in Sheng et al. [53], but could also be incorporated by using the transformed stress method of Yao et al. [68,69,70, 72].

All parameters of the sand model can be divided into three groups: (1) elasticity parameters (K₀, υ, and n), (2) CSL-related parameters (e_ref, λ, ξ, and ϕ_μ), and (3) interlocking-related parameters (A_d, k_p, n_p, and n_d).