Applications Of Two-dimensional Seepage Problem And Nonlinear Signorini Problem By The Localized Fundamental Solutions Of Method

In recent years,more and more attention has been paid to the capital construction in our country.Since a series of complex engineering problems caused by the capital construction need to be solved urgently,the investment generated in this regard continues to increase.In our daily life,the mathematical models of the most of engineering problems can be set as partial differential equations or ordinary differential equations with appropriate boundary conditions.However,due to some complex boundary conditions,the inhomogeneous properties of materials,irregular geometric shapes or other factors,it’s hard for us to find out the analytical solution of the problem by simple or direct derivation.The traditional experimental method requires high experimental cost and the validity of the experimental results is limited by the test facilities.Therefore,scholars take advantages of different numerical methods to obtain the approximate solutions for problems.Meanwhile,the development of modern computer hardware technology provides a vital support for the proposal and development of different numerical methods.Moreover,since more and more complex engineering problems are introduced,the limitations of traditional finite element method(FEM)in mesh generation continuously emerge.Scholars begin to explore a series of new numerical methods which do not require meshing.These methods can not only maintain high precision but also high efficiency,which are the so-called Meshless method.The localized method of fundamental solution(LMFS)is a stable and efficient new meshless method,which is developed from the traditional method of fundamental solution.The Localized method of fundamental solution retains the advantages of high precision of the traditional basic solution method and eliminates the matrix ill-conditioned problems caused by the full matrix.At the same time it has a wide application prospect.Localized method of fundamental solution(LMFS)is a stable and efficient new meshless method,which is developed from the traditional method of fundamental solutions(MFS).LMFS retains the advantages of the traditional MFS such as high precision.It can eliminate the matrix ill-conditioned problems caused by the full matrix,and has a wide range of application prospects.In this paper,the LMFS is applied to the study of solving two-dimensional nonlinear Signorini problems and steady-state seepage problems.Through combining the LMFS with relevant numerical techniques,the two types of problems mentioned above can be programmed to calculate by MATLAB.The electroplating problem,free dam problem,homogeneous isotropic steady-state seepage problem,and orthotropic steady-state seepage problem are given as examples in this paper.The results verify the feasibility and effectiveness of the LMFS.The research is described in detail as follows:(1)Firstly,the two-dimensional steady-state seepage problem is introduced to be solved.Since the different characteristics of the seepage medium,the hydraulic conductivity is also different.Thus,the treatment methods can be different as well.Two kinds of steady-state seepage problems are discussed by using the different relationship of hydraulic conductivity.When k_x=k_yand both are constants,the problem becomes a homogeneous isotropic seepage problem.When k_x and k_yboth are constants and not equal,the problem becomes an orthotropic steady-state seepage problem.We find out the fundamental solutions of the two problems and emerge a linear calculating system by discretizing governing equations and boundary conditions for numerical solutions.The accuracy solutions can verify the feasibility and effectiveness of the localized method of fundamental solution in solving this type of problems.(2)For the two-dimensional nonlinear Signorini problem,the ellipse boundary value problem in multi-connected geometries,the free dam problem and the electroplating problem are established.Then,the discrete solution is carried out by the the LMFS.Because the general elliptic boundary value problem is composed of Dirichlet boundary conditions and Neumann boundary conditions,the boundary conditions of Signorini problem which cannot be expressed as any linear formulation,are difficult to be programmed.The Signorini problem contains another special boundary condition,which is generally composed of two inequalities and given in a complementary form,so it’s impossible to solve the equation directly.(3)In order to solve this difficulty,this paper introduces the Fischer-Burmeister NCP-function to deal with the problem of transforming of the inequality of the Signorini boundary.Then,the original boundary condition given in the complementary form can be transformed into a nonlinear equation.What should be emphasized in the last,the built-in sfolve in MATLAB can be used for solving the nonlinear iteration in the process of calculation.(4)Finally,comparing all the numerical results above calculated in this paper with the numerical results in the previous work by other researchers,it can be found that the two kinds of numerical results can fit with each other very well.What’s more,on the premise of obtaining the accuracy results which share the similar order of magnitude with those in previous work,the number of collocation nodes required in this paper is much less and the efficiency is much more efficient than other numerical methods.In this paper,several examples are calculated by means of independent programming,and the feasibility and effectiveness of LMFS in solving steady-state seepage problems and nonlinear Signorini problems are verified.It provides new research thinking for solving more complex seepage problems and Signorini problems in the future.