Research On A Partition Systematic Methodology For Elastic Mechanics And First-order Weak Form Method

In this thesis,the research on integral form and superconvergence of finite element are summarized up.It is stated that the improvement of stress accuracy and some troubles of regular finite element solution.In conventional finite element method,the stress can be worked out through derivative of basic displacement function.The phenomenon of the order of magnitude decline will arise comparing the stress and nodal displacement in the accuracy and precision level.The problem how to improve and rise the accuracy of stress is central point of the researcher on finite element algorithm.It is established that partition systematic methodology for elastic mechanics.The theory is support for first-order weak form method.And,the first-order weak form of the elastic problem is derived,and then finite element programs are coded by using FreeFem++ software based on the weak form.The results of numerical calculation show that the accuracy of the stress and nodal displacement are on the same order.The main research work is as follows:(1)The basic conception and theory about weak form of the finite element are proposed.The FreeFem++ platform based on the weak form and it’s basic grammar and usage are introduced in detail.The physical problem that may be described through partial differential equation and corresponding weak form can be solved by using FreeFem++ software in finite element solution.The new algorithm suggested this thesis will be realized and verified rapidly provided that the software tool combines weak form.(2)The partition integral form and partition weak form of elastic mechanics are built based on differential form.The relationship and distinction of partition integral and partition weak form,conventional integral and conventional weak form are indicated.The duality property of the partition integral form on balancing of stresses and partition integral form on geometrical harmonize,the partition weak form on balancing of stresses and partition weak form on geometrical harmonize are proposed.The duality property reflect inner relation of partition integral form on balancing of stresses and partition integral form on geometrical harmonize.(3)The composition mode of integral form solution is analyzed.And,the definition of exact solution on finite element also is provided.The systematic methodology for elastic mechanics based on partition weak form is support for first-order finite element method in theory.In the derived finite element solution making use of the partition weak form,the continuous conditions of weighting function for displacement and stress on the partition element interface may are relaxed.It supply theoretical basis and loose range of choice for all kinds of the construction of non-conforming elements.(4)The rational method of building partition variational principle in elastic mechanics is presented.It is derived that various partition variational principle.Based on partition weak form,the partition virtual displacement equation and partition virtual stress equation are educed and the duality property of above two equation are proposed,the various partition variational principle are derived.Similarly,Based on partition integral form,the generalized partition virtual work equations are derived.And,The duality relation of the potential energy form and complementary energy form of the generalized partition virtual work equations is discussed.(5)On account of the virtual work equations of rigid body,take member bar system as an example,the new virtual work equations of rigid body considering internal force and relative displacement are set up.The new equations will unify the virtual work equations of rigid body with virtual work equations of transformed body.It is revealed that the duality property of the commutative transposed geometry matrix and equilibrium matrix in the virtual work equations of rigid body.(6)For the two-dimensional problems of elastic mechanics,this thesis deduce the corresponding second-order weak form and first-order weak form finite element method respectively.By means of some typical numerical examples,the solutions of two types algorithm conduct contrastive analysis.The calculation results of first-order weak form method show that the accuracy of the stress and nodal displacement are on the same order.(7)For the three-dimensional problems of elastic mechanics,this thesis deduce the corresponding second-order weak form and first-order weak form finite element method respectively.Taking some typical three-dimensional numerical problems as example,the results of the numerical calculation show that the accuracy of the stress and nodal displacement are also on the same order just as the previous two-dimensional problems.The first-order weak form finite element method that this thesis recommend can be extended and applied to the bend theory of thin plates,bend theory of medium plates,shell theory and computational fluid mechanics in finite element numerical calculation.The partition systematic methodology for elastic mechanics will accelerate the research on non-conforming elements and quasi-conforming elements.