Uncertainty assessment in hydro-mechanical-coupled analysis of saturated porous medium applying fuzzy finite element method

Abstract

The purpose of the present study was to develop a fuzzy finite element method, for uncertainty quantification of saturated soil properties on dynamic response of porous media, and also to discrete the coupled dynamic equations known as u-p hydro-mechanical equations. Input parameters included fuzzy numbers of Poisson’s ratio, Young’s modulus, and permeability coefficient as uncertain material of soil properties. Triangular membership functions were applied to obtain the intervals of input parameters in five membership grades, followed up by a minute examination of the effects of input parameters uncertainty on dynamic behavior of porous media. Calculations were for the optimized combinations of upper and lower bounds of input parameters to reveal soil response including displacement and pore water pressure via fuzzy numbers. Fuzzy analysis procedure was verified, and several numerical examples were analyzed by the developed method, including a dynamic analysis of elastic soil column and elastic foundation under ramp loading. Results indicated that the range of calculated displacements and pore pressure were dependent upon the number of fuzzy parameters and uncertainty of parameters within equations. Moreover, it was revealed that for the input variations looser sands were more sensitive than dense ones.

Keywords

fuzzy finite element method / saturated soil / hydro-mechanical coupled equations / coupled analysis / uncertainty analysis

Introduction

Current developments in efficient numerical tools and particularly uncertainty in evaluating influential parameters of porous medium as well as the concerned effects on soil response in various engineering problematic fields have directed the attention toward the idea in recent years [¹–³]. Due to soil heterogeneous and anisotropic nature, particularly various properties on sampling location and depth, achieving crisp values for soil properties can be unrealistic. Applying uncertainty and sensitivity analysis would provide solution and a reliable model in case of uncertainty in numerical modeling [¹–³]. Contributing uncertainty output and sensitivity analysis quantitatively measures uncertainties in different input parameters [³]. Modeling and analyzing uncertainty highly depends upon the representation of what is known as uncertain information. Formulating alternative strategy relies on the type and quantity of available information. Three classes of strategies were utilized to represent uncertainty in the last few years [⁴,⁵] including interval analysis [⁶,⁷], probabilistic approach [⁸–¹¹], and fuzzy theory [¹²–¹⁴]. In interval analysis, uncertain variables are denoted by a simple range or set, namely interval vector. Following a specified probability distribution, uncertain variables are described as random variables or processes and probabilistic approach will be applied [¹⁵]. Uncertainty in fuzzy theory is interpreted as designer’s and analyzer’s choice to utilize a particular value for uncertain variable. Accordingly, a preferred function is to describe desirability of applying different values within the same range. As expert judgment is often involved in parameter selection, the latter is more related to our context of geotechnical analysis [¹⁶]. Soil parameters with different numbers are taken from laboratory tests; therefore, final value parameters would expertly be selected. Engineering judgment errors are also detected in the analysis. Due to inhomogeneous and anisotropic structure of soil matrix, defining soil parameters via crisp numbers is obviously improbable. Results in one specific crisp value for uncertain input parameters cannot be the representative of the whole spectrum of possible results. Applying fuzzy arithmetic as a practical approach would be a good solution for these limitations and also applying fuzzy modeling approach and the alternate fuzzy numbers for the uncertainties of soil parameters, the concerned functions are expertly derived accordingly and results for soil environment analysis will be spectral considering the most critical conditions. The theory of fuzzy sets was first introduced by Zadeh [¹⁷] and since early 80s has been involved in engineering. In the beginning, it was applicable for engineering decision making and particularly designing rather than analyzing. Valliappan and Pham [¹⁶] were the firsts to utilize fuzzy finite element method (FFEM) in the analysis of engineering problems especially problems of foundation on elastic soil. Later they introduced elasto-plastic fuzzy finite element [¹⁸]. Cherki et al. [¹⁹] examined fuzzy behavior of mechanical systems with uncertain boundary conditions. Applying level optimization, Möller et al. [²⁰] studied fuzzy structural analysis. Hanss [²¹] published applications of fuzzy arithmetic in engineering problems. Interval analysis and fuzzy theory became powerful tools for real life applications and were widely used in structural analysis during the last decade [⁶,⁷,²²,²³]. FEM is described within fuzzy theory and is therefore known as Fuzzy Finite Element Method (FFEM). It aims to obtain a fuzzy description of an FE analysis, starting from fuzzy descriptions of all non-deterministic FE model parameters. FFEM was widely applied in different areas. Yang and Li [²⁴] introduced perturbation fuzzy finite element, which was later followed by Huang and Li [²⁵]. They compared the proposed approach with conventional fuzzy finite element method. Solving fuzzy sets was widely welcomed by some authors. Abbasbandy et al. [²⁶] deeply discussed a conjugate gradient method to solve fuzzy symmetric positive definite system of linear equation. Skalna et al. [²⁷] proposed three methods to solve systems of fuzzy equations in structural mechanics. Via illustrative examples, they described performances and advantages of presented methods. Mikaeilvand and Allahviranloo [²⁸,²⁹] proposed a novel method to find nonzero solutions for fully fuzzy linear systems. They used some numerical examples to illustrate their approach. Verhaeghe et al. [³⁰] introduced novel projection technique in fuzzy finite element analysis of structures and represent all parameters as trapezoidal fuzzy numbers and without any restriction on the coefficient matrix, Kumar and Bansal [³¹] proposed a new computational method to solve fully fuzzy linear system. Senthilkumar and Rajendran [³²] introduced a method to fully solve fuzzy linear systems with symmetric coefficient matrix. Via reanalysis technique of fuzzy finite element for static structural problems, Farkas et al. [³³] illustrated the concerned computational benefits and general applicability on a mid-sized plate problem. Based on high dimensional model representation, Balu and Rao [³⁴] presented a practical approach for analyzing structure response with fuzzy parameters. Babbar et al. [³⁵] examined some new numerical methods with triangular fuzzy numbers to solve a fully fuzzy linear system followed by some numerical examples. Assuming soil stiffness and damping ratio as uncertain parameters, Fujita et al. [¹²] recently presented robustness analysis of seismic pile response for a structure-pile-soil system. Considering uncertainties in the external load, material and geometric properties, Behera et al. [¹³] utilized fuzzy finite element method for non-probabilistic static analysis of imprecisely defined structures. There were limited studies on the analysis of soil problems with fuzzy approach and interaction effect in soil analysis was not fully covered [¹⁴,¹⁶,¹⁸]. The problem of modeling fluid flow in deformable porous medium has long been challenging and stems from the complex interactions between soil skeleton and pore fluid flow. It was initially formulated by Biot [³⁶] in 3D analysis of elastic soil matrix with Darcy pore fluid flow and followed by Zienkiewicz and Shiomi [³⁷] reformulation for the dynamic nonlinear analysis. Coupled equations were utilized as governing equations in different engineering problems [³⁸–⁴¹]. Moreover, Ghasemi et al. [³⁸,⁴⁰] applied related equations as design constraints for topology optimization of single and multi-material-based flexo-electric composites. Technical feasibility of utilizing hard rock for compressed air energy storage was examined using a coupled thermo-hydro-mechanical modeling of non-isothermal gas flow [⁴¹]. Badnava et al. [³⁹] via thermo-mechanical model studied brittle fracture and thermo-mechanical induced cracks. Finding an exact solution to these complicated problems is extremely challenging, therefore problems have been commonly solved numerically [⁴²–⁴⁷] and due to simplifications and assumptions related to values of input modeling parameters, results for this modeling approach are highly uncertain.

In the present study, for uncertainty quantification of soil properties on consolidation response of porous media a framework based on fuzzy finite element is developed. In this respect, dynamic coupled equations governing saturated porous medium known as u-p hydro-mechanical equations are analyzed by input fuzzy numbers including Poisson’s ratio, Young’s modulus, and permeability coefficient. Accordingly, for fuzzy analysis certain number of membership grades is first taken into account (five membership grades are reported here). For each membership grade and by triangular membership functions, possible ranges of parameters (intervals) are obtained. Secondly, to calculate upper and lower bounds of displacements and pore pressure, vertex method is applied and finite element calculations is performed for different combinations of upper and lower bounds of matrices as well. Calculations for all membership grades are repeated and finally result in a fuzzy number of displacements and pore pressure, which accordingly represent possible range in a certain membership grade (occurrence probability). Two different benchmark problems are analyzed as numerical examples. The first is estimating solution for elastic soil column and the later approximating elastic foundation response under ramp loading. Influences of fuzzy definition of Young’s modulus and Poisson’s ratio, together with permeability on the distribution of displacement and dissipation of excess pore water pressure are timely presented and discussed in full-depth.

Fuzzy sets

Fuzzy set is defined as a class of objects with a continuum of membership grades between the values of zero to one. A fuzzy set allows a gradual change from one class to another instead of an abrupt boundary as in an ordinary set. If M is considered as a universe set, set of A will be a fuzzy subset of M. It can be written as a set of ordered pairs as follows:

(1)A=\left\{\left[m,{{\displaystyle \mu }}_A\left(m\right),m\in M\right]\right\},

where m_A(m) is called membership grade of m and has a value in the closed interval of [0, 1]. This indicates probability level that the value of m belongs to A. The closer m_A(m) is to one, the more m belongs to A. The closer it is to zero, the less it belongs to A as well. This is to note that if [0, l] is replaced by the two-element set of {0, l}, then A can be regarded as an ordinary subset of M. For the sake of simplicity, the notion fuzzy set is utilized instead of fuzzy subset. Particular case of fuzzy sets is fuzzy numbers and the notion of fuzzy numbers is presented in different ways; one (which is presented here) would be to consider the quantification of a physical quantity in terms of a fuzzy set membership function [⁴⁸]. A fuzzy number such as A is a vertices subset of real numbers. Vertices means that for every real number a, b, c with a<c<b,

(2){{\displaystyle \mu }}_A\left(C\right)\ge \min \left({{\displaystyle \mu }}_A\left(a\right),{{\displaystyle \mu }}_A\left(b\right)\right).

It indicates that membership function of a fuzzy number consists of increasing and decreasing parts and there is only one member of the series such as z with a membership grade equals to one (m_A(z) = 1). Each fuzzy number is determined by its support, which is set as follows:

(3)supp\left(A\right)=\left\{x,{{\displaystyle \mu }}_A\left(x\right)>\mathrm{0}\right\}.

Convexity assumption (Eq. (2)) ensures that the support of fuzzy number is an interval and membership grade of a real number, which represents the probability of its occurrence. Level sets (in this study, intervals) represent different sets of numbers with least given probability [⁴⁹,⁵⁰].

Membership function

A fuzzy number is defined by membership function of m(x) and membership function is formed in various ways. Linear triangular membership function, illustrated in Fig. 1, is the simplest one and is defined as follows [¹⁸]:

(4)\mu \left(x\right)=\{\begin{array}{c}\hfill \mathrm{0},x\le l\text{'},\hfill \\ \hfill {\displaystyle \frac{x-l\text{'}}{m-l\text{'}}},l\text{'}\le x\le m,\hfill \\ \hfill {\displaystyle \frac{h\text{'}-x}{h\text{'}-m}},m\le x\le h\text{'},\hfill \\ \hfill \mathrm{0},x\le h\text{'},\hfill \end{array}

(5)l\text{'}=\{\begin{array}{c}\hfill m-\mathrm{2}\left(m-\mathrm{1}\right),m\ge \mathrm{2}\left(m-\mathrm{1}\right),\hfill \\ \hfill \mathrm{0},m\le \mathrm{2}\left(m-\mathrm{1}\right),\hfill \\ \hfill h\text{'}=m+\mathrm{2}\left(h-m\right),\hfill \end{array}

(6)h\text{'}=m+\mathrm{2}\left(h-m\right),

where l, m, and h are expert’s estimates for low, most likely, and high values of a parameter and l' and h' are the extreme low and high values of the parameters of l and h, respectively.

Mathematical model for interaction of pore fluid flow and soil skeleton

Biot [³⁶] proposed both equilibrium and continuity equations to analyze saturated porous medium. In Biot’s model governing equations for saturated porous medium with a single fluid phase, generally water, are formulated based on total equilibrium of soil-pore fluid mixture, mass balance of flow equation, concept of effective stress, constitutive model for soil behavior, and equilibrium equation for pore fluid which is called generalized Darcy’s Equation [³⁷]. Simply put, final governing equations involve two variables of u and p as follows. Details on these equations are given in Ref. [⁴²].

(7){{\displaystyle \sigma }}_{ij,j}+{{\displaystyle p\ddot{u}}}_i-{{\displaystyle \rho b}}_i=\mathrm{0},

(8){{\displaystyle \left[{{\displaystyle K}}_{ij}\left({{\displaystyle p}}_{,j}+{{\displaystyle \rho }}_{\mathrm{f}}{{\displaystyle \ddot{u}}}_i-{{\displaystyle \rho }}_{\mathrm{f}}{{\displaystyle b}}_j\right)\right]}}_{,i}+\alpha {{\displaystyle \dot{\epsilon }}}_{ii}+{\displaystyle \frac{\dot{p}}{C}}=\mathrm{0},

where {{\displaystyle \sigma }}_{ij} are total stresses, u is displacement vector of soil skeleton, and p is pore pressure. b_i is body force per unit mass, {{\displaystyle \rho }}_{\mathrm{f}}is fluid density, and \rho is the density of total composite which is defined by \rho ={{\displaystyle n\rho }}_{\mathrm{f}}+\left(\mathrm{1}-n\right){{\displaystyle \rho }}_{\mathrm{s}} in which n denotes porosity and {{\displaystyle \rho }}_{\mathrm{s}}the density of solid particles. {{\displaystyle K}}_{ij} is permeability per unit weight, {{\displaystyle \epsilon }}_{ij} are total strains and C is combined compressibility which is defined as C={\displaystyle \frac{n}{{{\displaystyle k}}_{\mathrm{f}}}}+{\displaystyle \frac{\left(\mathrm{1}-n\right)}{{{\displaystyle k}}_{\mathrm{s}}}}. k_s denotes bulk modulus of solid particles, and k_f bulk modulus of fluid. \alpha is dependent upon material type and is taken to be unite for soils. Equations (7) and (8) together form the u-p formulation, which must necessitates the solution in a coupled manner. Discrete form of coupled equation can be obtained as follows [⁴⁴]:

(9)\begin{array}{c}\hfill \left[\begin{array}{cc}\hfill {{\displaystyle \left[M\right]}}_{n+\mathrm{1}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \beta }}_{\mathrm{2}}{\mathrm{\Delta }t}^{\mathrm{2}}{{\displaystyle \left[{{\displaystyle k}}_m\right]}}_{n+\mathrm{1}}\hfill & \hfill -\mathrm{\Delta }t\theta {{\displaystyle \left[Q\right]}}_{n+\mathrm{1}}\hfill \\ \hfill {{\displaystyle {{\displaystyle \beta }}_{\mathrm{1}}{\mathrm{\Delta }t\left[Q\right]}^{\mathrm{T}}}}_{n+\mathrm{1}}\hfill & \hfill {{\displaystyle \left[S\right]}}_{n+\mathrm{1}}+\mathrm{\Delta }t\theta {{\displaystyle \left[{{\displaystyle k}}_{\mathrm{c}}\right]}}_{n+\mathrm{1}}\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \left\{{\mathrm{\Delta }u}_n\right\}\hfill \\ \hfill \left\{{\mathrm{\Delta }p}_n\right\}\hfill \end{array}\right\}=\hfill \\ \\ \hfill \left\{\begin{array}{c}\hfill {{\displaystyle \left\{f^{\mathrm{1}}\right\}}}_{n+\mathrm{1}}-{{\displaystyle \left\{f^{\mathrm{1}}\right\}}}_n+{{\displaystyle \left[Q\right]}}_{n+\mathrm{1}}\left\{p_n\right\}\mathrm{\Delta }t-{{\displaystyle \left[{{\displaystyle k}}_{\mathrm{m}}\right]}}_{n+\mathrm{1}}\left(\left\{u_n\right\}\mathrm{\Delta }t+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left\{u_n\right\}{\mathrm{\Delta }t}^{\mathrm{2}}\right)\hfill \\ \hfill {{\displaystyle \left\{f^{\mathrm{2}}\right\}}}_{n+\mathrm{1}}-{{\displaystyle \left\{f^{\mathrm{2}}\right\}}}_n-{{\displaystyle \left[{{\displaystyle k}}_{\mathrm{c}}\right]}}_{n+\mathrm{1}}\left\{p_n\right\}\mathrm{\Delta }t-{{\displaystyle \left[Q\right]}}_{n+\mathrm{1}}\left\{{u_n}_n\right\}{\mathrm{\Delta }t}^{\mathrm{2}}\hfill \end{array}\right\},\hfill \end{array}

where Dt is the time step. M, k_m, Q, S, and k_c are the mass, stiffness, coupling, compressibility, and permeability matrixes, respectively. f^{\mathrm{1}} and f^{\mathrm{2}} are nodal force vectors. {{\displaystyle \beta }}_{\mathrm{1}}, {{\displaystyle \beta }}_{\mathrm{2}}, and \theta are integration constants used in Newmark scheme. In this study, {{\displaystyle \beta }}_{\mathrm{1}}=\mathrm{0.6}, {{\displaystyle \beta }}_{\mathrm{2}}=\mathrm{0.605}, and \theta =\mathrm{0.6} are employed [⁴⁴]. Detailed explanation of discrete method together with definition of matrices and vectors are presented in Appendix A. With assuming linear behavior for soil leads to a linear system of equations, in Eq. (9). These sets of equations are solved by the developed FORTRAN finite element code in present study. The structure chart for incremental form of Biot analysis with fuzzy finite element method is illustrated in Fig. 2.

Fuzzy inputs

In governing equations, applying fuzzy numbers of elastic modulus and Poisson’s ratio for solid skeleton leads to fuzzy elastic matrix (D) and fuzzy stiffness matrix (k_m). Considering permeability coefficient as a fuzzy number for pore fluid, permeability matrix (k_c) will be a fuzzy number as well. Therefore, in matrix form of equation set (Eq. (9)), by two fuzzy matrices of k_m and k_c, and answers of equation set including pressure and acceleration rates as well as displacements and pressure will be regarded as fuzzy numbers. In addition to stiffness and permeability matrices on the right hand side of equations, displacement, velocity, acceleration, pressure, and pressure rate values of the previous time step are fuzzy numbers as well.

Fuzzy elastic matrix

In E, \upsilon fuzzy for plane strain problems (considering both Young’s modulus and Poisson’s ratio as fuzzy numbers), components of fuzzy elastic matrix to calculate lower bound of displacement at α membership grade are defined as follows [¹⁸]:

(10){{\displaystyle D^{\alpha }}}_{\mathrm{R}}={\displaystyle \frac{{{\displaystyle E^{\alpha }}}_{\mathrm{R}}}{\left(\mathrm{1}+{{\displaystyle \upsilon }}_{\mathrm{L}}^{\alpha }\right)\left(\mathrm{1}-{{\displaystyle \mathrm{2}\upsilon }}_{\mathrm{R}}^{\alpha }\right)}}\left[\begin{array}{ccc}\hfill \mathrm{1}-{{\displaystyle \upsilon }}_{\mathrm{L}}^{\alpha }\hfill & \hfill {{\displaystyle \upsilon }}_{\mathrm{L}}^{\alpha }\hfill & \hfill \mathrm{0}\hfill \\ \hfill {{\displaystyle \upsilon }}_{\mathrm{L}}^{\alpha }\hfill & \hfill \mathrm{1}-{{\displaystyle \upsilon }}_{\mathrm{L}}^{\alpha }\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill {\displaystyle \frac{\mathrm{1}-{{\displaystyle \mathrm{2}\upsilon }}_{\mathrm{L}}^{\alpha }}{\mathrm{2}}}\hfill \end{array}\right].

Also to calculate upper bound of displacement at α membership grade it becomes

(11){{\displaystyle D^{\alpha }}}_{\mathrm{L}}={\displaystyle \frac{{{\displaystyle E^{\alpha }}}_{\mathrm{R}}}{\left(\mathrm{1}+{{\displaystyle \upsilon }}_{\mathrm{R}}^{\alpha }\right)\left(\mathrm{1}-{{\displaystyle \mathrm{2}\upsilon }}_{\mathrm{L}}^{\alpha }\right)}}\left[\begin{array}{ccc}\hfill \mathrm{1}-{{\displaystyle \upsilon }}_{\mathrm{R}}^{\alpha }\hfill & \hfill {{\displaystyle \upsilon }}_{\mathrm{L}}^{\alpha }\hfill & \hfill \mathrm{0}\hfill \\ \hfill {{\displaystyle \upsilon }}_{\mathrm{L}}^{\alpha }\hfill & \hfill \mathrm{1}-{{\displaystyle \upsilon }}_{\mathrm{R}}^{\alpha }\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill {\displaystyle \frac{\mathrm{1}-{{\displaystyle \mathrm{2}\upsilon }}_{\mathrm{L}}^{\alpha }}{\mathrm{2}}}\hfill \end{array}\right].

Therefore, stiffness matrix (k_m) to calculate upper and lower bounds of displacement (u) and pore pressure (p) will be as follows:

(12){{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{uR}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{pR}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{L}}={{\displaystyle {\int \left[\mathit{B}\right]}^{\mathrm{T}}}}_{\mathrm{uL}}{{\displaystyle \left[\mathit{D}\right]}}_{\mathrm{L}}{{\displaystyle \left[\mathit{B}\right]}}_{\mathrm{uL}}\mathrm{d}\mathrm{\Omega },

(13){{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{uL}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{pL}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{R}}={{\displaystyle {\int \left[\mathit{B}\right]}^{\mathrm{T}}}}_{\mathrm{uR}}{{\displaystyle \left[\mathit{D}\right]}}_{\mathrm{R}}{{\displaystyle \left[\mathit{B}\right]}}_{\mathrm{uR}}\mathrm{d}\mathrm{\Omega }.

In k fuzzy (considering permeability coefficient as only fuzzy number), permeability matrix (k_c) to calculate upper and lower bounds of displacement (u) and pore pressure (p), the concerned functions will be as follows:

(14){{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{uL}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{pR}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{L}}={{\displaystyle {\int \left[\mathit{B}\right]}^{\mathrm{T}}}}_{\mathrm{pL}}{{\displaystyle \left[\mathit{k}\right]}}_{\mathrm{L}}{{\displaystyle \left[\mathit{B}\right]}}_{\mathrm{pL}}\mathrm{d}\mathrm{\Omega },

(15){{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{uR}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{pL}}={{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{R}}={{\displaystyle {\int \left[\mathit{B}\right]}^{\mathrm{T}}}}_{\mathrm{pR}}{{\displaystyle \left[\mathit{k}\right]}}_R{{\displaystyle \left[\mathit{B}\right]}}_{\mathrm{pR}}\mathrm{d}\mathrm{\Omega }.

In E, \upsilon, and k fuzzy (considering all, Young’s modulus, Poisson’s ratio, and permeability coefficient as fuzzy numbers), Eqs. (7) and (8) demand upper bound of stiffness matrix and lower bound of permeability matrix to calculate the lower bound of displacement, and vice versa for the upper bound of displacement. Moreover, upper bounds of stiffness and permeability matrices inversely calculate the lower bound of pore pressure. Equivalent nodal forces resulting from initial stresses to calculate lower and upper bounds of displacement and pressure in the right hand side of equations will be obtained by relevant stiffness and permeability matrices. For example in E, \upsilon, and k fuzzy (three parameters are fuzzy), right hand side of equations to calculate lower bound of displacement would be as follows:

{{\displaystyle \left\{{\mathit{F}}^{\mathrm{1}}\right\}}}_{\mathrm{uL}}=\left\{{\mathrm{\Delta }\mathit{f}}^{\mathrm{1}}\right\}+{{\displaystyle \left[\mathit{Q}\right]}}_{n+\mathrm{1}}{{\displaystyle \left\{{\dot{\overline{\mathit{p}}}}_n\right\}}}_{\mathrm{uL}}\mathrm{\Delta }t

(16)-{{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{R}}\left({{\displaystyle \left\{{\dot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{uL}}\mathrm{\Delta }t+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \left\{{\ddot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{uL}}{\mathrm{\Delta }t}^{\mathrm{2}}\right),

(17){{\displaystyle \left\{{\mathit{F}}^{\mathrm{2}}\right\}}}_{\mathrm{uL}}=-\left\{{\mathrm{\Delta }\mathit{f}}^{\mathrm{2}}\right\}+{{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{L}}{{\displaystyle \left\{{\dot{\overline{\mathit{p}}}}_n\right\}}}_{\mathrm{uL}}\mathrm{\Delta }t+{{\displaystyle \left[\mathit{Q}\right]}}_{n+\mathrm{1}}{{\displaystyle \left\{{\ddot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{uL}}\mathrm{\Delta }t,

Equivalent nodal forces for the upper bound of the displacement are:

{{\displaystyle \left\{{\mathit{F}}^{\mathrm{1}}\right\}}}_{\mathrm{uR}}=\left\{{\mathrm{\Delta }\mathit{f}}^{\mathrm{2}}\right\}+{{\displaystyle \left[\mathit{Q}\right]}}_{n+\mathrm{1}}{{\displaystyle \left\{{\dot{\overline{\mathit{p}}}}_n\right\}}}_{\mathrm{uR}}\mathrm{\Delta }t

(18)-{{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_{\mathrm{L}}\left({{\displaystyle \left\{{\dot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{uR}}\mathrm{\Delta }t+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \left\{{\ddot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{uR}}{\mathrm{\Delta }t}^{\mathrm{2}}\right),

(19){{\displaystyle \left\{{\mathit{F}}^{\mathrm{2}}\right\}}}_{\mathrm{uR}}=-\left\{{\mathrm{\Delta }\mathit{f}}^{\mathrm{2}}\right\}+{{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{R}}{{\displaystyle \left\{{\dot{\overline{\mathit{p}}}}_n\right\}}}_{\mathrm{uR}}\mathrm{\Delta }t+{{\displaystyle \left[\mathit{Q}\right]}}_{n+\mathrm{1}}{{\displaystyle \left\{{\ddot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{uR}}\mathrm{\Delta }t.

Also for fuzzy number of pore pressure (PP) it is expressed as

{{\displaystyle \left\{{\mathit{F}}^{\mathrm{1}}\right\}}}_{\mathrm{pL}}=\left\{{\mathrm{\Delta }\mathit{f}}^{\mathrm{1}}\right\}+{{\displaystyle \left[\mathit{Q}\right]}}_{n+\mathrm{1}}{{\displaystyle \left\{{\dot{\overline{\mathit{p}}}}_n\right\}}}_{\mathrm{pL}}\mathrm{\Delta }t

(20)-{{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{m}}\right]}}_R\left({{\displaystyle \left\{{\dot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{pL}}\mathrm{\Delta }t+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \left\{{\ddot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{pL}}{\mathrm{\Delta }t}^{\mathrm{2}}\right),

(21){{\displaystyle \left\{{\mathit{F}}^{\mathrm{2}}\right\}}}_{\mathrm{pL}}=-\left\{{\mathrm{\Delta }\mathit{f}}^{\mathrm{2}}\right\}+{{\displaystyle \left[{{\displaystyle \mathit{k}}}_{\mathrm{c}}\right]}}_{\mathrm{R}}{{\displaystyle \left\{{\dot{\overline{\mathit{p}}}}_n\right\}}}_{\mathrm{pL}}\mathrm{\Delta }t+{{\displaystyle \left[\mathit{Q}\right]}}_{n+\mathrm{1}}{{\displaystyle \left\{{\ddot{\overline{\mathit{u}}}}_n\right\}}}_{\mathrm{pL}}\mathrm{\Delta }t.

Equivalent nodal forces for the upper bound of pressure are given by replacing subtitles of pL with pR, and L with R. In all cases and for each membership grade, equation answers are obtained via \left[{{\displaystyle \left({{\displaystyle \mathrm{\Delta }\ddot{\overline{\mathit{u}}}}}_n\right)}}_{\mathrm{L}},{{\displaystyle \left({{\displaystyle \mathrm{\Delta }\ddot{\overline{\mathit{u}}}}}_n\right)}}_{\mathrm{R}}\right],[{{\displaystyle \left({{\displaystyle \mathrm{\Delta }\dot{\overline{\mathit{p}}}}}_n\right)}}_{\mathrm{L}},{{\displaystyle \left({{\displaystyle \mathrm{\Delta }\dot{\overline{\mathit{p}}}}}_n\right)}}_{\mathrm{R}}]. Applying these values, quantities of relevant acceleration, velocity, displacement, and pressure rate are calculated. In Eqs. (10)–(21), L and R subtitles stand for lower and upper bounds, respectively. Communications between fuzzy and deterministic solvers are illustrated in Fig. 3.

Numerical simulation results

Numerical examples of elastic soil column and elastic foundation are presented in order to illustrate the application of fuzzy finite element method in the analysis of deformable porous medium and are simulated by a plane strain representation. The first example verifies the model. Calculations were in membership grades of 0.5 to 1 to avoid Poisson’s ratio of 0.5.

Elastic soil column

To illustrate the accuracy and versatility of proposed finite element method in deformable porous medium, many researchers studied examples of elastic soil column [⁴⁷]. A fully saturated soil column subjected to a surface step loading was analyzed by FFEM in this study and the concerned results of vertical displacement and pore pressure at membership grade of one was utilized for model verification. The column has a width of w = 1.0 m and a height of H= 30 m, it is subjected to a surface step loading of 1 \mathrm{kN}/{\mathrm{m}}^2 applied in 0.1 s at the top level. Drainage was allowed only via the top surface of column and problem was modeled as a saturated soil column under plane strain condition. Boundary conditions for displacement field were included all nodes, which were horizontally constrained and at the bottom level were fixed with no vertical movement. Atmospheric pressure existed at the top level and impermeable boundaries were imposed at lateral and bottom surfaces. Geometry and boundary conditions of soil column are illustrated in Fig. 4. Material properties are given in Table 1. The problem includes two regions with a horizontal material interface as shown in Fig. 4. Mesh consists of 10 eight and four node quadrilateral elements for displacement and pressure, respectively. Time step is set to 0.05 s and FEM mesh is given in this figure as well. By considering the general properties of loose and dense sands in FFEM model, two existing layers were analyzed and accordingly input fuzzy numbers were chosen based on reported data for possible values of Poisson’s ratio, Young’s modulus, and permeability coefficient in different soil literatures (see Tables 2 [⁵¹] and 3 [⁵²]). In fuzzy analysis, corresponding fuzzy numbers of Poisson’s ratio (v), Young’s modulus (E), and permeability coefficient (k) were used instead of their crisp values. To maintain the shape of output functions, linear triangular shape functions were applied for all fuzzy parameters. Membership functions for input fuzzy parameters are illustrated in Fig. 5. As they are listed in Table1, membership functions are obtained using parameters of the least amount l, the most likely value m, and the maximum value h. Minimum and maximum values were chosen in such a way that the first layer determined the loose sand and the second layer the dense to medium dense sand. Fuzzy finite element analysis of the sample problem at membership grade of one (see Fig. 4), deterministic analysis, was conducted using most likely values (m) of soil parameters, deterministic soil parameters, to verify the established numerical model. Figure 6 illustrates variation of pore water pressure and increase of settlement during time at different nodes within soil column. Results for deterministic analysis at membership grade of one indicated a relevant harmony with those reported by Khoie and Haghighat [⁴⁷]. Figure 7 depicts time histories calculated by FFEM for different membership grades in pore pressure (PP) and displacements at various nodes of soil column. Results within time revealed that in the beginning which increases by depth (node 6:0.1 s, node 16:1 s, node 36:5 s, node 46:13 s) pore water pressure was in its highest value and the lower and upper bounds coincided with each other.

Initial ground settlement was almost zero. During consolidation, ground settlement gradually increased via dissipation of excess pore water pressure, and difference between lower and upper bounds increased by time. Relevant time in pore water pressure was dependent upon the studied level after reaching its maximum value. It decreased when both lower and upper bounds reached zero. As a result of D_elastic, variation of Young’s modulus and Poisson’s ratio affect displacement and pressure values by stiffness matrix (k_m), and permeability matrix (k_c) also has impact variation of permeability coefficient. Per numerical simulation in all three cases of analysis by E,v fuzzy, only E fuzzy and only v fuzzy, applying k_m (min) pinpointed maximum displacement and minimum pressure. Applying k_m (max) determined minimum displacement and maximum pressures as well. By increasing drainage distance and strengthening interaction effect, pressure had more influence upon displacement and subsequently overcame the effect of stiffness matrix. Therefore, the pattern of lower bound and upper bound calculation were replaced with each other and in two nodes of 36 and 46, k_m (min) determined minimum displacement and pressure while k_m (max) reported maximum displacement and pressure. The least changes in response were related to those nodes located in the longest drainage distance. By increasing drainage distance, for other nodes located in less dense materials with greater pore volume, interaction became more in coupled equations. So in all cases of analysis including both E and v fuzzy, only E fuzzy and only v fuzzy, pore pressure was affected by changes in soil parameters. As depth increased, wider range of solutions was obtained for pore pressure. This meant that the effect of soil stiffness variation on its permeability increased by depth. Displacement was inverted and the concerned range decreased by depth. For example due to the greater drainage distance for node 11 than node 6 in soil column in Figs. 8 and 9, displacement was less influenced and pressure was more affected by change in soil parameters respectively. As it is shown in Fig. 7, displacement at two nodes of 46 and 36 (located in the second layer of soil column, the densest materials) for different membership grades, has not significantly changed in early times. Due to low permeability, high drainage distance and limited pore sizes, interaction particularly in early times was very high in coupled equations. Unlike displacement, change in soil parameters hada significant impact on pore pressure. While there was an increase in time and excess pore pressure disappeared, interaction between fluid and solid skeleton decreased and change in soil parameters affected displacement more and more. Increase in the elastic modulus specified value of E = 60 MPa, denser sand, pore volume reached a minimum possible value and variation in vertical displacement due to decrease in pore volume almost stopped. According to Fig. 7, these nodes upper bound pore pressure and displacement had no significant change and were almost matched with the most likely amount (α = 1). Figures 8 and 9 illustrate displacements and pore pressures fuzzy numbers in different nodes of soil column within 200 s after loading. As mentioned earlier in this study, permeability reduction increased pressure and decreased displacement and the concerned model represented by the researcher vividly reflected what was necessarily expected from soil behavior. Therefore, each of these qualitative terms for interaction effect on displacement and pore pressure were quantitatively explicable by fuzzy finite element analysis in any particular time or location.

Elastic foundation

Second example was an elastic foundation, which was subjected to a surface step loading with three internal sub-domains. Foundation problem was one of the most popular examples utilized by various researchers to capture shear band localization [⁵³], effect of sub-domain interfaces [⁴⁷] as well as the analysis of displacements [¹⁸]. In this study and via fuzzy analysis, sensitivity level of this foundation dynamic analysis to input soil parameters was examined. With this purpose, step loading of 350 kN/m² was applied in 0.1 s on the surface of fully saturated soil foundation. Pore pressure was assumed to be p = 0 at the top of the foundation and foundation dimension 30 m in 40 m, respectively. Figure 10 illustrates geometry and boundary conditions and Fig. 11 shows fuzzy numbers of Poisson’s ratio (v), Young’s modulus (E), and permeability coefficient (k). Input fuzzy variables support was selected perfectly symmetric in two regions of one and two. Similar to the example of soil column, the eight and four node quadrilateral elements were applied for displacement and pressure, respectively. Time step was set to 0.05 s. Dynamic analysis of a fully saturated soil foundation was conducted by three fuzzy input parameters. Table 4 illustrates the material properties of foundation for three regions. Maximum and minimum variations of pore pressure and vertical displacement within time were plotted at different points of foundation in Figs. 12 and 13, respectively. In spite of symmetric variation of upper and lower bounds of input fuzzy parameters, analysis results for displacement and pore pressures were asymmetric over time. Depend upon the studied level, asymmetry of pore pressure gradually increased within time and reached maximum value during consolidation.

While variation of input parameters was high, the effect in less membership grades was more obvious (see Fig. 12) similar to pattern of variations in pore pressure values. Asymmetry in vertical displacement was almost constant over time and more vivid in less membership grades (see Fig. 13). Figure 14 illustrates displacement and pore pressure fuzzy numbers in elastic foundation within 100 s after loading. To find the source of asymmetry, each parameter and pairs as the only fuzzy input parameters were analyzed and followed by in-depth analysis of concerned correlated effects (see Figs. 14 and 15). According to Figs. 14 and 15, asymmetry in displacements is mainly due to Young’s modulus as an input fuzzy parameter. Asymmetry in pore pressures was also due to permeability coefficient as an input fuzzy parameter (see Fig. 15(c)). In the analysis with only E as a fuzzy parameter (E fuzzy in the above figures) and despite symmetric change in the only existing fuzzy input parameter of Young’s modulus, pore pressure and displacement output fuzzy numbers were asymmetric. It clearly indicated that the upper bound changes for displacement and pore pressure were more than lower bound changes. Vertical displacement asymmetry in this case was almost constant by depth yet horizontal displacement and pore pressure asymmetry decreased by depth (see Fig. 15). This is to mention that asymmetry, relative to deterministic solution, was used to describe difference between variation percentage value in lower and upper bounds. For example when variation percentage of upper and lower bounds was 15.3 and -12.1, asymmetry value was 3.2. As stated earlier in this study, reduction in Young’s modulus reduced stiffness and affected the upper bounds of pressure and displacement and enlargement in Young’s modulus affected lower bounds, respectively. While the upper bound was more than the lower bound, reduction in Young’s modulus had more impact on pressure and displacement than the time it was enlarged. In sandy soil, greater Young’s modulus described denser sand and in higher densities, soil environment pure size was less followed by low interaction. In less dense sands, interaction was more and analysis was more sensitive to the value of input variable. Therefore, in looser sands slight variations in input parameters led to greater changes in outputs. For example in membership grade of 0.9 with three input parameters including 5% variation in Poisson’s ratio (v), 10% variation in both Young’s modulus (E), and permeability coefficient (k) affected displacements almost by 15% in looser part and by 11% in denser part (see Figs. 15(a) and 15(b). Pore pressure was dependent upon the studied depth. In lower depth, it varied by 23% in both two parts but in high depth, it varied by 20% in looser part and by 18% in denser part (see Fig. 15(c)). For the effect of permeability fuzzy number in lower depth, interaction effects were added as depth increased. Accordingly, in looser environments the importance of fuzzy analysis was more evident. Based on the interaction equations, pore pressure was directly correlated with depth and inversely correlated with displacement. Although horizontal displacement asymmetry decreased (see Fig. 15(a)) and vertical displacement asymmetry was almost constant (see Fig. 15(b)), pressure asymmetry increased as depth went up (see Fig. 15(c)). Figure 15 quantitatively illustrates the four nodes of elastic foundation at membership grade of 0.9 within 100 s with comparing the results of displacements and pore water pressure at 1 and 100 s after loading, interaction effect is clearly justifiable. During the early times in the foundation (see Figs. 16(a) and 16(b)), pore pressure (PP) was highly constant in value and there was no significant change between minimum and maximum values. With the passage of time and due to drainage and transferring load into the solid part of soil, the effect of changing soil parameters became more evident (see Figs. 17(a) and (b)). Pore pressure values decreased and there was a difference between minimum and maximum values. According to Figs. 16, 17(c), and 17(d) and within 1 s after loading, both location and minimum, maximum vertical displacements values were different. After 100 s, only response values changed.

Figures 16, 17(e), and 17(f) illustrate that during earlier times, horizontal displacements were just under the loading part and gradually developed all over the foundation. Interestingly, displacement pattern progressed from loaded area to other parts of foundation. Soil displacement value and stabilization in looser part with maximum output was more than denser part with minimum output. Porosity of these two parts influenced the interaction and the concerned analysis indicated that fuzzy model could quantitatively predict the expected soil behavior over time and after loading. It provided reliable solution intervals for better judgment on the possibility and location of cracks by settlements, liquefaction, and induced pore pressure during dynamic analysis.

Conclusions

Within finite element as a practical method of analyzing complicated problems in geotechnical engineering including liquefaction and cracking in soil structures [⁵⁴–⁵⁹] via earthquake, input soil properties are generally imprecise and impractical to be described with crisp numbers. However, during dynamic analysis of porous medium, reasonable treatment as fuzzy numbers reflect realistic estimate for the value of displacements and induced pore pressure. In this study, a fuzzy finite element model was proposed to analyze the dynamic coupled response of saturated porous medium via treating soil properties as fuzzy numbers. Determining quantitatively, how certain variation in input parameters affected solution. Results of the current study showed that the proposed method was promising. This new modeling framework is suitable for many geotechnical problems where uncertainties are due to insufficient data or imprecise information. One of distinctive advantages of fuzzy set approach for real situations is the involvement of expert knowledge. In addition to fuzzy parameters analysis, membership grades are helpful for the purpose of engineering design. In fuzzy finite element soil analysis, interaction between soil skeleton and pore water, as a basic principle governing the porous medium, must be considered. Interaction between different domains severely affects soil response particularly in dynamic loading. The concerned effect is depicted in different soil densities by fuzzy finite element method.

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