The First-order Finite Element Solution Based On The Freefem++

In the usual finite element method, the approximation of displacement fieldfunction is supposed firstly. Usually infinite points are defined in the every unit. Acontinuous body is divided into finite element which is thought of as an ensembleonly connect through these points each other. The value of these points is consideredas basic unknown quantities. Then we use the variation method to obtain the finiteelement equation which can work out the the value of these points. The value of allstress components is obtain by displacement derivative, which caused the accuracy ofstress decreases by an order of magnitude when compared with the accuracy ofdisplacement. These problems bring new attention to the accuracy of stress, andproduce important topic that how to let the accuracy of stress be with order to theaccuracy of displacement.Based on previous researches, a new kind of method has been discovered. Andthe method is the first-order finite element solution which based on the FreeFEM++.The method enabled the accuracy of stress to be with order to the accuracy ofdisplacement. The main achievements are as follows:(1)By carding the data and materials about the topic, the paper firstly introducesthe resolution method of elastic mechanics and development level about the globalerror of the finite element solution on home and abroad. After that, the paperintroduces the method of getting finite element equation in details and sums up theoutcome about improve the accuracy of stress that the former researchers hadmentioned. On the basis of New Systematic Methodology for Elasticity to establishthe solving theory based on the weak form, which provides theoretical basis forresearching the new method in the paper.(2)The paper researches and use freefem++ software of finite element method. Bycalculating and analysising the example, the paper introduces the basic grammar andthe calculation principles of the software in details.(3)Basing on previous researches, the Second-order weak form and the First-orderweak form were derived for two-dimensional elasticity. The accuracy of the methodsabove is analyzed comparatively through typical examples. Among them the Second--order weak form corresponding to the usual finite element method and the First-orderweak form corresponding to the method of multi-variate variational form. We canrealized that the first-order finite element solution enabled the accuracy of stress to be with order to the accuracy of displacement directly. At last, the paper extend themethod to the three-dimensional question successfully.