A Stabilization Procedure for Soil-water Coupled Problems Using the Mesh-free Method

Transcription

1 The 12 th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 28 Goa, India A Stabilization Procedure for Soil-ater Coupled Problems Using the Mesh-free Method T. Shibata Dept. of Civil and Environmental Engineering, Matsue National College of Technology, Shimane, Japan A. Murakami Graduate School of Environmental Science, Okayama University, Okayama, Japan Keyords: Mesh-free method, soil-ater coupled problem, stabilization procedure ABSTRACT: The development of stability problems related to classical mixed methods has recently been observed. In this study, a soil-ater coupled boundary-value problem, one type of stability problem, is presented using the Element-free Galerkin Method (EFG Method). In this soil-ater coupled problem, anomalous behavior appears in the pressure field unless stabilization techniques are used. The remedy to such numerical instability has generally been to adopt a higher interpolation order for the displacements than for the pore pressure. As an alternative, hoever, an added stabilization term is incorporated into the equilibrium equation. The advantages of this stabilization procedure are as follos: (1) The interpolation order for the pore pressure is the same as that for the displacements. Therefore, the interpolation functions in the pore pressure field do not reduce the accuracy of the numerical results. (2) The stabilization term consists of first derivatives. The first derivatives of the interpolation functions for the EFG Method are smooth, and therefore, the solutions for pore pressure are accurate.in order to validate the above stabilization technique, some numerical results are given. It can be seen from the results that a good convergence is obtained ith this stabilization term. 1 Introduction The development of numerical computation technologies has enabled a variety of engineering problems to be solved and has brought about remarkable progress in recent decades. Among the related findings, meshless and/or mesh-free methods in particular have been applied to some problems for hich the usual finite element method is ineffective in dealing ith significant mesh distortion brought about by large deformations, crack groth, and moving discontinuities. Various meshless and/or mesh-free methods have been used for geotechnical problems, instead of the finite element method, to overcome the above-mentioned difficulties. Consolidation phenomena have been analyzed by means of EFGM (Modaressi et al., 1998)(Nogami et al., 24)(Murakami et al., 25)(Wang et al., 26), the point/radial point interpolation method (PIM/RPIM) (Wang et al., 21)(Wang et al., 22), the local RPIM (Wang et al., 25), RKPM (Chen et al., 21)(Zhang et al., 25), and the natural neighbor method (Cai et al., 25), the transient response of saturated soil has been dealt ith under cyclic loading by means of EFGM (Karim et al., 22)(Sato et al., 26), ave-induced seabed response and instability have been examined by EFGM (Wang et al., 27) and RPIM (Wang et al., 24), slip lines have been modeled by geological materials using EFGM (Rabczuk et al., 26), and a Bayesian inverse analysis has been carried out in conjunction ith the meshless local Petrov-Galerkin method (Sheu, 26). Hoever, unless certain requirements are met in dealing ith soil-ater coupled problems for the finite element computation, based on the coupled formulation becoming ill-conditioned, numerical instabilities ill occur (Chapelle et al., 1993). In order to overcome these eaknesses, several strategies have been proposed (Pastor et al., 1999). For example, as a necessary condition for stability, the interpolation degree of the displacement field must be higher than that of the pore pressure field. An alternative means of stabilization as also proposed based on the Simo-Rifai enhanced strain method hich even allos an equal order of interpolation degree for both variables. Hoever, these strategies are not directly applicable to meshless/mesh-free methods, because all the nodal points simultaneously have the same degree of freedom for both the displacement field and the pore pressure field, and no information beteen the element and the nodes can be utilized. The purpose of this paper is to present a stabilization methodology for the mesh-free analysis of soil-ater coupled problems by incorporating the stabilizing term into the eak form. The folloing sections deal ith descriptions of the formulation, including the stabilization term. In Section 3, to applications of the strategy to 64

2 soil-ater coupled problems are analyzed, one being the saturated soil column test appearing in Mira et al.(23), to demonstrate the effectiveness of the strategy, and the other being the foundation behavior under displacement-controlled condition, for hich the feasibility of theanalysis ill be thoroughly discussed hile Table 1. Order of the interpolation function in FEM Number Pressure Order Displacement :Pressure :Displacement Figure 1. Rearrangement and allocation of background cells to cover the changing domain comparing the finite element solutions. Conclusions ill follo in Section 4. 2 The Stabilization term The governing equations for soil-ater coupled problems ith boundary conditions and initial conditions are given as follos: div S & δ v dv + ρ v v δ v d S = (1) V t Γ u ( ) V V Γ Γ ( tr ) δhdv + v gradδhdv qδhds β ( p p ) D δhds = (2) here S & t is the nominal stress rate, v is the velocity, v is the boundary value of the velocity, D is the stretching, h is the total head, v is the velocity of the pore ater, q is the discharge per unit area ith units of length per time, β is Lagrange multiplier, p is the pore ater pressure, and p is the boundary value of the pore ater pressure. As previously mentioned, mixed displacement-pressure formulations (e.g., finite element methods) produce locking phenomena in the pressure field unless stabilization techniques are used. The remedy for such numerical instability has generally been to adopt a higher interpolation order for the displacements than for the pore pressure. As an alternative, hoever, an added stabilization term is incorporated. It is shon in this study that the instability can be eliminated by adding the stabilization term to the continuity of pore ater (Equation (2)) hich consists of the square of the pore ater pressure of the first derivatives. q h + δ ( tr D) δhdv + v gradδh dv qδh ds β ( p p ) V q h 2 α ( p ) dv = V Γ Γ V δh ds (3) here α is the stability parameter, p is the differentiate p ith respect to z, and z is the vertical axis. In this study, the soil column test created by Zienkieicz et al. is performed in order to examine the numerical stability of this procedure, and the values for the pore ater pressure are illustrated along the vertical axis of the soil column. In order to smooth the values for the pore ater pressure along the vertical axis ith the continuity condition of the pore ater, the stabilization term is differentiated ith respect to z. The to advantages of this stabilization procedure are as follos: (1) The interpolation order for the pore pressure is the same as that for the displacements, namely, a loer interpolation order is not adopted for the pore 65

3 pressure. Therefore, the interpolation functions in the pore pressure field do not reduce the accuracy of the numerical results. (2) Table 1 summarizes the order of the interpolation functions for the pressure field and for the displacement field in FEM. The first derivatives of the interpolation functions in the pore pressure field are the zero order or the first order, as shon Table 1. Therefore, accuracy in the numerical results cannot be obtained. With EFGM, hoever, the interpolation functions are derived by the MLS approximant and a linearbased polynomial is used, in other ords, the resultant interpolation functions are smooth. The eak forms are integrated using the MLS interpolation functions in space and employing the explicit time scheme in time. In the manipulation of the stiffness matrix, a numerical integration is performed on the Table 2. Material parameters Compression index λ.11 Selling index κ.4 Critical state parameter M 1.42 Poisson s ratio ν.333 Initial void ratio e.83 Initial volume ratio v = 1+ e 1.83 Initial consolidation stress (kpa) p 294 z = 3 p = u = 3 m z = u x = u = z 1 m q p z p x z = z u = x = x Figure 2. Saturated column test 3 Depth ( m ) 2 1 k= ( cm/sec ) Pore ater pressure ( Pa ) [ 1 5 ] ithout stabilization term α =.1 α =.1 α =.1 Figure 3. Effect of the dimensionless parameter background cells. The background cells, hich intersect or make contact ith the boundary, are divided into four finer cells. The cells are then rearranged during the computation, according to changes in the domain, as shon in Figure 1. 3 Numerical Examples 3.1 The saturated column soil test For soil-ater coupled problems, numerical instabilities are often encountered at the initial stage under undrained conditions unless stabilization techniques are performed. In this chapter, the numerical stability of an EFG computation is examined using the proposed stabilization term described in the last chapter. Let s consider the 1D problem analyzed by Zienkieicz et al.(1986) in hich the saturated soil column test in Figure 2 uses the material parameters listed in Table 2. Here, a sand permeability of cm/s is employed. A model discretized by 62 nodal points is adopted. Linear-based polynomials are applied for the interpolation functions of the EFG Method. The functions have the same order for both the displacements and the pore pressure. The eight function is a quartic spline type of eight function and its radius of support is 1.. Here, 5 5 Gaussian points are used. The background cells are 1.m ithin the domain and.5m outside of the domain. The scale factor, hich is defined as the magnification of the support diameter to the side length of the square background cell, is 1.5. This study employs penalty methods to apply the boundary conditions, and the value of the penalty factor is In order to solve the stiffness equation, the forard difference is 66

4 adopted. Figure 3 shos the effect of the proposed stabilization term and dimensionless parameter α on the numerical profile of the pore pressure just after loading under undrained conditions, for hich a time difference of Δ t =.1 and a value of permeability of k = cm/s are adopted. There are 62 nodal points and the eight function is a quartic spline. From Figure 3, it can be concluded that the numerical solution has been improved in the case here the value of the dimensionless parameter is.1. In subsequent examinations, a permeability of cm/s and a time difference of.1 are adopted. 2. Here, the numerical instability is caused from the large difference of the value in the stiffness matrix in beteen displacements and pore ater pressures. Generally, the values of stiffness matrix in pore ater pressure are smaller than in displacements. Hoever, if the loer order interpolation function in pore ater pressures is adopted, the values of stiffness matrix in pore ater pressures become large, so that the difference of the values in stiffness matrix becomes small. Similarly, if the stabilization term is considered, the values of stiffness matrix Loading plate p = λ =.11 κ =.4 M = 1.42 ν =.333 e =.83 v = 1+ e =1.83 p = 294 (kpa) 5. m 4.m 2.m 4.m 1.m (a) Geometry and boundary conditions (b) Initial collocation of the nodal points Figure 4. Sketch of the problem [kpa] (a) Contours of the pore pressure (b) Contours of the normalized strain (c) Displacement distributions (d) Collocation of the nodal points Figure 5. Numerical results ithout stabilization term 67

5 in pore ater pressure also become large. Therefore, the stabilization term quells the anomalous pore pressure behavior because of the small difference. 3.2 Foundation Problem Subjected to Strip Loading In order to examine the numerical availability of the stabilization procedure to EFGM, a 2D soil-ater coupled problem is solved in relation to a soft soil foundation. In this problem, the p& in Equation (1) is the differentiate p ith respect to x and z, ith x being the horizontal axis and z being the vertical axis. The geometry and the boundary conditions are given in Figure 4(a), hile the initial collocation of the nodal points is shon in Figure 4(b). Vertical displacements are applied at the top face to simulate loading on the foundation, hile the [kpa] (a) Contours of the pore pressure (b) Contours of the normalized strain (c) Displacement distributions (d) Collocation of the nodal points Figure 6. Numerical results ith stabilization term Applied footing stress [MPa].2.1 Prandtl's solution EFG solution Settlement [m] Figure 7. Comparison EFG solution ith Prandtl s solution transverse direction is restrained. Thus, the boundary conditions under the loading surface are given by the settlements, and the other locations on the upper boundary are stress free. The bottom boundary is fixed in all directions, hile the side boundaries are fixed in only the horizontal direction, such that vertical displacement is alloed. Hydrostatic pressure is used here for the initial conditions in the pore pressure field. The model is generated ith 1,326 nodal points, in other ords, 26 vertical nodal points and 51 horizontal nodal points. The size of background cells is.2m, and the value of the dimensionless parameter α is same as the previous analysis. Figures 5 and 6 present the numerical results ithout and ith the stabilization term, respectively, in hich (a), (b), (c) and (d) sho the contours of the pore ater pressure, the contours of the normalized strain, the displacement distributions and collocation of the nodal points, respectively. In these results, the settlements 68

6 under the loading surface are.4m in Figures 5 and.4m in Figures 6, respectively. The normalized strain measure is given as t here ε = Ddt (Yatomi et al., 1989). T ( ) ε = tr εε (2) Spurious oscillations arise in the pore pressure field, as can be seen in Figure 5(a). In contrast to the results using the stabilization procedure, high values are obtained for the pore pressure and the normalized strain in Figures 5(a) and (b), respectively. In particular, anomalous behavior appears on left side in these figures. Moreover, the directions of the displacement vectors are disorderly in Figure 5(c) because of the interaction beteen the pore pressure field and the displacement field. If stabilization techniques are used, hoever, no oscillations appear in the solution, as observed in Figure 6. Figure 6(a) accurately displays the rise in the pore ater pressure belo the loading surface. In Figure 6(b), the prominently localized zones of the normalized strain occur just beneath the edge of the loading surface. The deformed pattern in Figure 6(c) is very similar to the classical slip line solution obtained by Prandtl. The shear bands are recognized as the localized deformation. Figure 7 compares the EFG solution ith Prandtl s solution. The Prandtl s solution q f is expressed as q f =5.14 c u (3) here c u is the undrained shear strength. From this figure, it is revealed that the EFG solution approaches Prandtl s solution. This result shos that the EFG method ith stabilizing procedure is very efficient for solving problems of computational geomechanics. 4 Conclusion In this paper, e have proposed a stabilization method for soil-ater coupled problems using the Element-free Galerkin Method. A stabilization term has been presented by the addition of a continuity condition. The proposed stabilization procedure has the folloing to characteristics: (1) The interpolation order for the pore pressure field is the same as that for the displacements, in other ords, a loer interpolation order for the pore pressure is not adopted. (2) The stabilization term consists of first derivatives in hich the interpolation functions are smooth because of the MLS approximation. The saturated column test and the foundation loading problem have been solved using the stabilization procedure. Numerical examples have shon that the stabilization method can indeed quell the anomalous pore pressure behavior. 5 References Modaressi H, Aubert P., 1998, Element-free Galerkin method for deforming multiphase porous media, International Journal for Numerical Methods Engineering, 42, Nogami T, Wang W, Wang JG., 24, Numerical method for consolidation analysis of lumpy clay fillings ith meshless method, Soils and Foundations, 44(1), Murakami A, Setsuyasu T, Arimoto S., 25, Mesh-free method for soil-ater coupled problem ithin finite strain and its numerical validity, Soils and Foundations, 45(2). Wang ZL, Li YC., 26, Analysis of factors influencing the solution of the consolidation problem by using an element-free Galerkin method, Computers & Geosciences, 32, Wang JG, Liu GR, Wu YG., 21, A point interpolation method for simulating dissipation process of consolidation, Computer Methods in Applied Mechanics and Engineering, 19, Wang JG, Liu GR, Lin P., 22, Numerical analysis of Biot s consolidation process by radial point interpolation method, International Journal of Solids and Structures, 39, Wang JG, Yan L, Liu GR., 25, A local radial point interpolation method for dissipation process of excess pore ater pressure, International Journal of Numerical Methods for Heat & Fluid Flo, 15(6), Chen JS, Wu CT, Chi L, Huck F., 21, A meshfree method for geotechnical materials, Journal of Engineering Mechanics, 127, Zhang JF, Zhang WP, Zheng Y., 25, A meshfree method and its applications to elasto-plastic problems, Journal of Zhejiang University SCIENCE, 6A(2), Cai YC, Zhu HH, Xia CC., 25, Meshless natural neighbor method and its application to complex geotechnical engineering, Yanshilixue Yu Gongcheng Xuebao/Chinese Journal of Rock Mechanics and Engineering, 24(11), Karim MR, Nogami T, Wang JG., 22, Analysis of transient response of saturated porous elastic soil under cyclic loading using element-free Galerkin method, International Journal of Solids and Structures, 39, Sato T, Matsumaru T., 26, Numerical simulation of liquefaction and flo process using mesh free method, Journal of Geotechnical Engineering, 813, (in Japanese). 69

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