Research On Superconvergent Recovery Algorithm For Fe Analysis On Two-Dimensional Problems

Differential equations have various forms,and their analytical solutions are relatively complex and have low applicability.Only a few differential equations with special forms have analytical solutions,so analytical solutions can't meet the need of solving complex problems in practical engineering.Therefore,numerical solutions have been extensively studied.For continuous system problems,the finite element(FE)method based on the weighted residual method and the variational principle is the most widely used.There's a trade-off between efficiency and accuracy of the FE method.Therefore,scholars at home and abroad have made a lot of research on the post-processing process based on the natural superconvergent properties.Inspired by the idea of dimensionality reduction by finite element method of lines,a p-type superconvergent recovery strategy which has two steps with boundary first and element second is proposed based on superconvergent properties of two-dimensional mesh nodal displacements in FE solutions.A row of adjacent elements with common opposite edges is taken out as a sub-domain.A local boundary value problem of the original partial differential equations on it which the true solution approximately satisfies is established by setting FE solutions on each element's other opposite edges as Dirichlet boundary conditions.By increasing the element order along the elements' opposite edges direction unidirectionally,the local boundary value problem is solved by the FE method to obtain the superconvergent displacement solution on the opposite edges of each element in this sub-domain.The superconvergent solution on the other opposite edges of elements can be obtained similarly with another sub-domain with respect to the edges to be recovered.Based on the recovered edge solutions,each element domain is taken out,and the original problem on it is solved using a higher order Lagrange element with the superconvergent solution on its edges set as Dirichlet boundary conditions.Thus the superconvergent solution of the whole domain can be obtained.In this paper,two-dimensional Poisson equation,plane problems of elasticity and thin plate problems with small deflection are taken as research objects,and the algorithm is successfully applied to the boundary value problem of second-order and fourth-order partial differential equations.Then the rule of convergence order is briefly demonstrated for the FE analysis with Lagrange elements on Poisson equations.On the basis of the above p-type superconvergent recovery method,a local algorithm is further proposed in this paper.The specific strategy is to take out several elements near the edge to be repaired rather than the area extending to the boundary as the solution domain in the first step of the process.This change allows the algorithm to realize the repair of local areas with poor accuracy of the FE solutions such as the boundary,which enhances the flexibility of the application.And through numerous experiments,the empirical value of the local elements' number is given.By using FE solutions on a series of sub-domains to obtain the superconvergent solution for two-dimensional problems,this algorithm inherits the advantages of the one-dimensional p-type superconvergent algorithm.The scale of post-processing computation is small,and the accuracy and quality of solutions are significantly improved.Moreover,the programming is simple and the method is easy to popularize.Therefore,the algorithm in this paper has the value of further research.