Analysis Of Material Nonlinearity Problems By Using Manifold Method Based On Independent Covers

Many of the problems in engineering descend to solving partial differential equation or partial differential equations.However,there is only a small part of the partial differential equations that can find the analytical solution.With the improvement of computer computing ability,numerical computing methods have become the mainstream.Among various numerical methods,the current theory and algorithm that have been relatively well developed is the finite element method,and its position in the numerical calculation is almost unshakable.However,the finite element method requires mesh shape to be as regular as possible,so the preprocessing is heavy work;The nodes of adjacent elements are required to coincide,so it is not convenient to increase the mesh density;the calculation process requires more degrees of freedom,and the stress calculation accuracy is relatively low.In 2011,Su Haidong proposed a new method,that is,manifold method based on independent covers.The method divides the physical field into independent covers of arbitrary shape,in which the real field function is approximated by using complete series.Independent covers are connected by using strips,with the connection of each series by using weight functions to form a continuous global approximation function.This method has clear concepts,arbitrary mesh divisions,high computing accuracy and good development prospects.By now,the manifold method based on independent covers has not been applied to material nonlinear analysis.Based on the program of the elastic analysis using the new method,referring to the previous theory and procedure of the finite elment method of elastoplastic calculation,the purpose of calculating the material nonlinear problem using the new method is achieved.Referring to the numerical calculation of finite element method,the fundamental equation and the incremental form of the manifold method based on independent covers are discussed by the principle of minimum potential energy and variational method.Referring to the D.R.J.Owen’s theory,method and program of elastoplastic calculation by using finite element method,in this paper the FORTRAN program for elastic-plastic plane stress and plane strain problems of the new method is writen,with 4 yield criteria(Tresca,Von Mises,Mohr-Coulomb,Drucker-Prager),and the linear and nonlinear hardening of the material,and four different incremental iteration methods(Newton method,Modified Newton Method and two hybrid methods).We first calculate a standard cantilever calculation example,and compare the results with the finite element method using large commercial software ANSYS to verify the reliability of the program.Then we use this program to calculate an engineering case.Considering the concrete elastic-plastic constitutive model of nonlinear hardening,the yield states under compression of a prestress lining of a water transport tunnel under the condition of underthickness are analyzed.The displacements,equivalent stresses and equivalent plastic strains before and after water supply are given.Since the underthick part is the key area of the yield,the local mesh density of these part is increased to obtain more accurate and reasonable results.The calculation shows that the prestress applied under underthickness conditions directly causes the structure to yield locally,so the construction of the actual works should vigorously avoid the underthickness of lining.This example also illustrates the practical value of the new method and this program.This paper avoids the complexity of preprocessing,the inconvenience of mesh density transition,and more degrees of freedom for analyzing material nonlinear problems when using the finite element method.It shows the advantages of arbitrary mesh division,arbitrary mesh refinement,small calculation scale and high calculation accuracy in the new method.It also lays a good foundation for future study of multiple nonlinear analysis of material nonlinearity,geometric nonlinearity,and contact nonlinearity by using manifold method based on independent covers.