Using Fuzzy FEM in Dynamic Coupled Analysis of Saturated Porous Media

In the present study, a fuzzy finite element model is developed to apply uncertainty of soil parameters on dynamic behavior of coupled saturated porous media. The interaction problem in the analysis of elastic soil matrix with Darcy pore fluid flow which is formulated by Biot is one of the complicated problems and its exact solution is so difficult, therefore it can be solved numerically. Finite Element Method is one of the numerical methods to approximate the dynamic solution of these problems and for convenient approximation of solution, model parameters need to be precisely known. On the other hand due to inhomogeneous and anisotropic structure of soil matrix, it is not possible to define the soil parameters with the crisp numbers. Consequently results that are obtained only one specific crisp value for an uncertain parameter cannot be representative for the whole spectrum of the possible results. To solve this limitation, application of fuzzy arithmetic proves to be a practical approach. For this purpose uncertainties in the soil parameters are taken in to account by the fuzzy numbers and shape function of input fuzzy numbers are derived from experimental data. In this study the coupled equations governing saturated porous media which are known as u-p equations, are solved by fuzzification of input parameters. For this purpose input parameters, the Poisson's ratio and modulus of elasticity, are treated as fuzzy numbers. To fuzzify a parameter, a certain number of degrees of membership are considered and by using fuzzy rules for each degree of membership a range of parameters are obtained which has lower and upper bound and the calculations are done for this upper and lower bounds. As a numerical example, problem of elastic soil column consist of two layers, loose and compacted sand, is analyzed. For solving this problem a finite element Fortran code has been developed and verified. For verification of developed Fortran code, the support of input fuzzy numbers was adopted in a way that the most likely amount of input parameters (m) be equal to the crisp input numbers that had been used by previous studies. The results indicated solution at α=1(probability of one hundred percent, which is the definite solution) is in good agreement with results of literatures. For other degrees of membership, the problem was solved with two constitutive D matrix for each α level to compute lower and upper bounds of output for that level. At the end, displacement and excess pore water pressure of sandy soil column which are produced under rapid loading are reported as fuzzy numbers. It means that it can be seen if Poisson's ratio and modulus of elasticity increase or decrease what will happen to displacement and excess pore water pressure. The results showed that coupling, change of input parameters of soil skeleton influence pore water pressure too. And as time increase this effect can be seen well. Also increase drainage distance causes a decreasing in the interaction between soil skeleton and pore water and this effect decreases by depth. Obtained result show that the number of fuzzy parameters (uncertain parameters) in equations increase the range of answers.