Conjugate Gradient Method For The Numerical Solution Of Elastic Contact

In this paper,we mainly consider the elastic contact problem.Two elastic ob-jects will appear elastic deformation when they are in contact and the contact surface will produce the corresponding stress and deformation displacement.When they are in equilibrium,the elastic potential energy is minimal.According to the principle of min-imum potential energy,we can turn the elastic problem into a large-scale optimization problem in mathematics by deriving variational equations.When the elastic boundary conditions and load conditions become more com-plicated,the boundary value problem of partial differential equation often can not get the analytical solution,so the classical solution of elastic mechanics have some limitations.To this end,a lot of research conducted on the numerical solution of e-lastic mechanics,for example,the difference method,variational method have been published.In this paper,we mainly study the conjugate gradient method which is used to solve the convex quadratic programming problem derived from elastic contact.Exist-ing algorithms are hindered based on conjugate gradient method to solve the convex quadratic programming problem obtained by elastic contact.For example,it is too cautious for the method in dealing with the effective set,which leads sometimes the algorithms very complicated and not easily convergent;the conjugate gradient method,in the iterative process,needs to be restarted many times,thus the convergence of the algorithm is greatly reduced.In this paper,a new algorithm model is proposed based on the conjugate gradient method and the effective set method.By using the simple projection,the iterative point can be ensured in the feasible region by the interception of variables when the effective set varies.We also use the Polak-Bibiere formula to avoid restarting the conjugate gradient algorithm.In the large-scale elastic contact problem,a large number of numerical results show the effectiveness of the algorithm.