Research On Super-convergent Algorithm In FEM Based On Improved Displacement Mode

Research on super-convergent computation in the field of Finite ElementMethod(FEM) has been a hot and difficult subject in recent years. Based on thecomparison of improved displacement mode and conventional displacement mode ofGalerkin equations, the exact conditions for the establishment of Galerkin equation isderived. Based on the premise of this paper, improved displacement mode isproposed,the displacement mode of high-order finite element solution is expressedwith a conventional finite element solution, the new displacement model isconstructed with the sum of the displacement model of conventional finite elementand the high-order displacement model of finite element solution, combined with thethinking of multi-scale method and the analytical formulas of superconvergencecomputing, a new pretreatment method of superconvergence calculation is proposed.The method does not need any artificial polish, for one-dimensional FEM, thedisplacement and the derivative accuracy of nodes and elements have reached h4order. The stresses so calculated are automatically in equilibrium across elements andare exact at free ends or edges, the implementation is simple, the work of calculationincrease very small.The Finite Element Method of Lines(FEMOL) is a continuous developmentsubject,At home and abroad study of superconvergence calculation on FEMOL islittle.Used the semi-analytical nature of FEMOL,the superconvergence calculation ofimproved displacement mode is also applied, to two-dimensional FEMOL and similarconclusions have been made.The main work of this paper is as follows:1. Based on the comparison of improved displacement mode and conventionaldisplacement mode of Galerkin equations, the exact conditions for the establishmentof Galerkin equation is derived. The association of displacement mode and the bubblefunction is demonstrated. The combination program of liner element and bubblefunction is proposed. Using the condition of precise establishment, by the analysisorder of magnitude, retaining the main impact item of bubble function, the analyticalexpression solving of bubble function is avoided. The theoretical basis of the analysisof the displacement mode is provided.2. Taking one-dimensional C0problem as a model problem, a pretreatment method of superconvergence calculation based on improved displacement mode isproposed for FEM. for improving the accuracy of the solution and reducing theresidual of balance equation, the higher-order trial functions is added at thefoundation of the original trial function, for approximate elements, according to theinternal equilibrium of unit,a complete set of formulas for super-convergentdisplacements and derivatives at any point on an element are also derived. Theelement stiffness matrix is derived using the integral form.3. A pretreatment method of superconvergence calculation based on improveddisplacement mode is successfully extended to other problems in one-dimensionalFEM,such as one-dimensional C1problem and the for second order non-self-adjointboundary-value problem, now that, the method can be used for not only self-adjointoperator problem but also non-self-adjoint operator problem. The problem ofone-dimensional n-order can be sloved by this method.4. A pretreatment method of superconvergence calculation based on improveddisplacement mode is successfully extended to Poisson equation problems intwo-dimensional FEMOL, based on linear shape function the modified ordinarydifferential equations of FEMOL is derived using the variational form. The result ofexamples show that the displacement and the derivative accuracy of nodes andelements has been greatly improved.Using super-convergent stresses and displacements, the associated errorestimation and self-adaptive solution strategy may also be developed in the future.