Special Element Methods With Applications To Geotechnical Engineering

The advancement of computer technology promotes the rapid development of numerical analysis methods.In recent years,meshless method,discrete element method,numerical manifold method and other numerical methods have also been developed and applied to a certain extent.But finite element methods have been used widely because of its sound theoretical and best universality.For numerical methods,there are two main points of view.One is to develop new numerical methods to meet special needs,and another is to extend the finite element method to meet special needs.This paper agrees with the second viewpoint.We should extend the finite element method to meet more special needs.In order to distinguish the traditional finite element method,the two methods studied in this paper are collectively called special element method.The first method is used to deal with the problem of fracture with finite element method.The traditional finite element methods need to set the location of the crack and to remesh during the process of calculation in dealing with the problem of fracture,but the extended finite element method(XFEM)can avoid remeshing during the process of calculation by using the discontinuous functions to simulate the displacement,and it can make cracks independent of the mesh.Another method breaks through the limitation of element shape and makes it possible to deal with more complex problems accurately.Now the finite element method use triangular elements and quadrilateral elements widely in two dimension because it can not calculate the explicit trial functions and it will have difficult dealing with the problems of complicated boundary conditions and complicated materials.The virtual element method(VEM)is on the basis of finite element method and finite difference method.It combine the degrees of freedom with the trial functions inside the element,although not necessarily polynomials by using the space mapping.It can extend the finite element method to any polygonal element and it do not need the explicit trial functions during the process of calculation.In this paper,the above two special numerical methods will be studied.At present,there are many studies on XFEM,but the research on VEM is still very deficient.This paper will focus on VEM.The main contents and conclusions of this paper are as follows:(1)Basic theories of XFEM are studied and introduced and the XFEM procedure of the linear elasticity problems are writed in FORTRAN based on the knowledge of fracture mechanics.The accuracy of the procedure is proved by a numerical example about the fracture of two dimension plate.From the example,we can see that XFEM do not need to remesh during the process of calculation,it can make simpler to analyze the discontinuous problems of fracture,so we suggest using XFEM to deal with the problems of cracking and propagation of cracks.(2)Basic theories of VEM are studied and the method to solve problems with VEM are introduced.The procedures to solve the problems of elastic ground and flow are writed and their accuracy are proved by numerical examples.From the numerical examples we can see that:a)The VEM procedure of linear elasticity is proved by numerical examples,and the orthogonal rule suggested based on the different displacement model is proved right.b)VEM can be used to analyze the problems of flow,and it can capture the steady field of flow accurately along with the change of water level.VEM will have better accuracy with the increase of element number,c)When we use polygonal element,the VEM can have same accuracy using less elements compared with FEM.For some geotechnical problems,we suggest using VEM to calculate and analyze,it can have same accuracy and reduce the amount of calculation and save the storage space,and it will improve the efficiency of analysis.