Stability And Strain Localization Analysis Of Geotechnical Problems Based On Second-order Cone Programming Optimized Finite Element Method

Geotechnical stability and strain localization analysis have always been the important research topic in geotechnical engineering.In the traditional computational framework,complex equilibrium iteration and stress integration algorithms are required for incremental elastoplastic analysis,which increases the complexity of program development.Based on the second-order cone programming and finite element theory,the incremental elastoplastic analysis of the classical continuum and the Cosserat continuum can be cast into a form of second-order cone programming(SOCP)which can be solved by the standard mathematical optimization solver(e.g.MOSEK).Second-order cone programming optimized finite element method for classical continuum(i.e.FEM-SOCP)and second-order cone programming optimized finite element method for Cosserat continuum(i.e.Cos FEM-SOCP)are proposed,respectively,and applied to geotechnical stability and strain localization analysis under plane strain conditions.The main research work includes the following aspects:(1)Based on the Hellinger-Reissner mixed variational principle and finite element method,the elastoplastic analysis for geotechnical problems can be cast into a second-order cone programming problem under the finite element framework,and the second-order cone programming optimized finite element method for classical continuum is established(i.e.FEM-SOCP),which can be solved by standard mathematical optimization solvers.The FEM-SOCP method is verified based on a ground example under the flexible foundation.It can be observed from the numerical analysis that the load-displacement curve calculated by FEM-SOCP method is basically consistent with the traditional FEM method,and the correctness of FEM-SOCP program and the codes are verified;(2)The FEM-SOCP method is applied to the geotechnical stability and deformation analysis.Combining the FEM-SOCP method with the strength reduction technique,the shear strength reduction finite element method based on the second-order cone programming theory(i.e.SSRFEM-SOCP)is proposed.Through a series of slope examples,it is found that the plastic zone obtained by the new method is smoother than the traditional SSRFEM method.For the geotechnical stability analysis with non-associated plasticity,considering that the results calculated by the original Davis formula are relatively conservative,especially for the case of high internal friction angle and low dilatancy angle.Based on the relationship between stress circle and strength envelope,a modified Davis formula is proposed based on the original Davis formula,then applied to the slope stability analysis,and a reasonable parameter value for the modified Davis approach is given(i.e.?=0.3).Since only the associated plasticity problem can be cast into the second-order cone programming,a time-discrete method with automatic error control is proposed for the iterative scheme of approximating the non-associated plastic problems with a series of associated plastic problems.Based on numerical investigations,the time step factor with the range ?= 0.1?0.4 generally may lead to an accurate solution with an optimized efficiency;(3)In the framework of second-order cone programming optimized finite element method,the incremental elastoplastic governing equation for the Cosserat continuum can be cast into the standard SOCP problem,and a second-order cone programming optimized finite element method for Cosserat continuum(i.e.Cos FEM-SOCP)is established,and the final solution format of the Cos FEM-SOCP method under force and displacement conditions are given,respectively.The Cos FEM-SOCP method can be solved using standard mathematical solvers,avoiding complex nonlinear iterations and stress integration algorithms.The elastic stress concentration problem around the circular hole is investigated in order to verify the correctness of the Cos FEM-SOCP program.The stress concentration factor calculated by the Cos FEM-SOCP method is basically consistent with the analytical solution and other numerical solutions,and the relative error does not exceed 1.6%,and the correctness of Cos FEM-SOCP program is verified;(4)The Cos FEM-SOCP method is applied to the geotechnical strain localization and stability analysis.Based on the elastic-perfectly plastic ground problem,the elastoplastic degradation solution of the Cosserat continuum is consistent with the traditional FEM results.Through the uniaxial compression test,the effect of shear modulus ratio,internal characteristic length and softening modulus on the shear band and load-displacement curve is investigated;besides,the selection relationship between internal characteristic length and the finite element size is investigated.Based on the Cosserat continuum theory,the mesh sensitivity problem due to strain softening can be overcome by using appropriate internal characteristic lengths.Finally,the stability and strain localization of soil slope are studied based on Cos FEM-SOCP method.The results show that the shear zone of the slope toe is sensitive with different finite mesh based on the classical continuum theory,while mesh-independent results can be obtained based on the Cosserat continuum theory;in addition,the safety factor of the classical continuum is relative conversive,and the safety factor based on Cosserat continuum will increase as the internal characteristic length increases.