Adaptive Fem Analysis For Systems Of First Order Odes Based On EEP Super-convergent Method

The numerical solution of ODEs (Ordinary Differential Equations) plays an important role in the modern mechanics and engineering computing. Any initial value problems and boundary value problems of high order ODEs can be converted into equivalent first-order ODE systems, so the numerical solution of first-order ODE systems is especially fundamental.For Galerkin FEM analysis of first-order ODE systems, the present dissertation proposed an EEP (Element Energy Projection) method for computation of super-convergent results, and the self-adaptive strategy based on the EEP method was applied to the numerical solution of the first-order ODE systems. This method could reliably and efficiently obtain a solution satisfying the specified tolerance pointwise. This research paved the way for developing a new ODE Solver based on FEM. In addition, taking the cylindrical shells under axisymmetric load as an example, the self-adaptive strategy based on EEP method was applied to the direct computation of mixed order ODEs.The main work of this dissertation is as follows:1. The EEP method was proposed for the super-convergent computation in Galerkin FEM for first-order ODE systems. Mathematical analysis and a number of given numerical experiments show that, the simple and convenient EEP condensed scheme and simplified scheme could respectively obtain super-convergent solutions with optimal super-convergence order and strong super-convergence order at any point of the domain.2. Based on the EEP method, a self-adaptive strategy was constructed for Galerkin FEM analysis of first-order ODE systems. This strategy proves to be straightforward in theory and simple in implementation, and could obtain a final adaptive mesh on which the conventional Galerkin FEM solutions satisfy the specified tolerance pointwise. Representative numerical examples show that the present adaptive method has many advantages over the well-known ODE solver COLSYS in the aspects of error control, computation efficiency, etc. 3. A shell element with optimal super-convergent nodal displacements was constructed first, and then EEP formulation for the super-convergent computation in FEM for the mixed order ODEs of cylindrical shells under axisymmetric load was established. Typical numerical examples show that the EEP condensed scheme is capable of producing optimal super-convergence for both displacements and nodal stresses at any point of the domain, and the simplyfied scheme could raise the convergence order by two over that of the FEM solutions.4. Based on EEP method, a self-adaptive strategy was constructed for cylindrical shells under axisymmetric load. For this kind of mixed order ODEs, the strategy could also get solutions reliably and efficiently, which satisfy the specified tolerance at any point of the whole domain.