Adaptive FEM Analysis Of Second Order Odes Of Femol Based On EEP Super-convergent Method

Adaptive analysis is a modern goal for various numerical methods. The mesh generation and precision controlling of which are both finished automatically by procedure without artifical work, and the reliability of which is guaranteed, adaptive analysis is also an important part of the modern FEM (Finite Element Method) research. For the boundary value problems (BVPs) denoted by second order linear self-adjoint ordinary differential equations (ODEs) derived from Finite Element Method of Lines (FEMOL), The present dissertation proposed strategies of self-adaptive FEM analysis based on Element Energy Projection (EEP) method, and the preliminary solver was formed. The work promoted the flexibility and applicability of EEP super-convergent method, also provided the foundation for complete adaptive analysis of two-dimensional structural problems based on EEP method. The main work of this dissertation is as follows:1. Taking FEMOL ODEs as the model problem, the exact element theory in FEM analysis was established and two EEP super-convergent schemes, simplified form and condensed form, were proposed for approximate elements. The simplified form gains certain degree of super-convergence while the condensed form is capable of producing optimal h2 m super-convergence for both displacements and derivatives at any point on an element. The strategy is a successful extension of EEP method to second order ODEs of FEMOL with all advantages in single ODE problems being reserved.2. Based on two EEP super-convergent solutions with high accuracy and high shape-fidelity, and the error-averging method for mesh refinement, the present dissertation proposed self-adaptive strategies for second order ODEs of FEMOL. Between these strategies, single-step method is basic for mesh genaration. With standard error estimation and mesh refinement, the single-step method gives little redundant final mesh, and is easy to implement on parametric elements. Multi-step method uses alternative mesh generation program, that is, adaptive piecewise polynomial interpolation associated with simplified super-convergent solutions and quasi-FEM interpolation associated with condensed super-convergent solutions, and can give multi-elements in a self-adaptive step. Multi-step method is more efficient in aspect of mesh generation, but has a dependence on accuracy and shape-fidelity of super-convergent solutions on initial meshes.3. Adaptive analysis of second order ODEs based on EEP method was applied to solve various FEMOL problems. The preliminary adaptive FEM solver based on EEP method was formed for second order ODEs of FEMOL, and succeeded in solving ODEs derived from Poisson equation, plane elasticity and mindlin-reissner plate bending problems. The strategy proposed in this dissertation is beyond the well-known ODE solver COLSYS in aspects of error control and reliability. The adaptive analysis based on EEP simplified super-convergent scheme is efficient, and the mesh of it has little redundance, but the error control is not extremely strict. The adaptive analysis based on EEP condensed super-convergent scheme is much reliable, so the solutions of each line can pointwice satisfy the maximum norm error control, but sometimes the mesh had redundant elements.