The Generalized Modal Method In Finite Element Design With Its Application

Thin-walled structures are widely used in aviation,aerospace,navigation and automobile engineering fields due to their high strength-over-weight and stiffness-over-weight ratios.With the development of computer technology and the increased demand of modern industry,the numerical simulation of thin-walled structures using finite element methods has become an important technical tool in fine engineering design,which can effectively reduce the cost of structural test and speed up the design process.For the large-scale engineering problems with thin-walled structures,the finite element model are generally discretized by beam elements,shell elements and solid elements to achieve the goal of efficient numerical structure analysis.Since the thin-walled structures are described in the form of three-dimensional solids in the CAD modeling process,the use of beam elements and shell elements requires additional geometric transformation of the initial geometrical description of the structure,which significantly increases the time and cost of finite element modeling.Furthermore,the connection between the beam element,shell element and solid element is very complicated.Therefore,the solid-type elements,including solid element and solid-shell element,have been a hot area of research in the recent decades.However,the performance of the existing solid-shell elements cannot fully meet the requirements of high efficiency modeling and high precision analysis of thin wall structure.To this end,the study of new high-precision and high-efficiency finite element method has important academic research and engineering application value.Aiming at the technical defects in modern finite element theory and method,this paper presents a generalized modal element method,which produces a number of high precision element formulations for thin-walled plates and carries out preliminary application research of the proposed elements.The main work of the paper is as follows:(1)Four common modal construction methods including analytical method,assumed displacement method,traditional finite element method and approximation method are proposed.The concept of modal local coordinate system is proposed to ensure that the elements derived from analytic method and the assumed displacement method satisfy the requirement of frame invariance.The convergence condition of element derived from GMEM is deduced,and the definition of modal completeness is given to evaluate the mechanical properties of the element constructed by the GMEM.(2)The basic deformation modes and unphysical modes of eight-node hexahedral elements are constructed by using the analytic method and assumed displacement method,respectively.As US-MEM8 S is difficult to apply in structural frequency and linear buckling analysis,a symmetric hexahedral element S-MEM8 S and its geometry stiffness matrix are proposed based on the displacement functions.(3)Based on GMEM,two six-node solid shell elements US-CTRIA3 and US-OTS3 are proposed,in which the in-plane modes are derived from analytical method and six-node linear isoparametric solid element and the out-of-plane modes are constructed by using triangular shell elements(CTRIA3 and OTS3).Using modal transformation method,two symmetric solid-shell elements S-CTRIA3 and S-OTS3 are derived from US-CTRIA3 and US-OTS3.By using the transformation of the nodal DOF between shell and solid-shell elements,the geometry stiffness of S-OTS3 is derived from OTS3 shell element.(4)Two eight-node solid shell elements US-CQUAD4 and US-QTS4 are proposed,in which the in-plane modes are derived from analytical method and assumed displacement mehtod and the out-of-plane modes are constructed by using quadrilateral shell elements(CQUAD4 and QTS4).As US-CQUAD4 and US-QTS4 cannot be applied in the frequency and linear buckling analysis,two symmetric solid-shell elements S-CTRIA3 and S-OTS3 are proposed by using modal transformation method.Furthermore,the geometry stiffness of S-QTS4 is derived from QTS4 shell element by using the transformation of the nodal DOF between shell and solid-shell elements.(5)Based on the linear superposition of element deformation modes,the compatible constraints of nodal displacement are derived for the analysis of finite element model with non-matching grids.Specifically,the compatibale displacement constraint matrixes of S-MEM8 S and CQUAD4 are deduced and verified that the classical multi-point constraint methods applied in NASTRAN and ABAQUS are not suitable for the analysis of complex deformation with non-matching grids.To tackle with this problem,a flexible multipoint constraint mehthod for non-matching grids is proposed by constructing the load transfomation matrix and the compatible displacement constraint matrix respectively.(6)As different selection of mass matrix may result in different inertia loads in classical inertia relief method,we deduce the sufficient condition for developing element mass matrix that can accurately describe the structural inertia load.Furthermore,since the mass matrix of beam element in NASTRAN and ABAQUS cannot describe the inertia load accurately,a new mass matrix of beam element is derived based on the proposed sufficient condition.Finally,a modified inertia relief method is proposd to save the computational time and memory that occupied by classical inertia relief method.