More generalized hybrid variational principle and corresponding finite element model

According to recent studies of the generalized variational principle by Professor Chien Weizang, the more generalized hybrid variational principle for finite element method is given, from which a new kind of the generalized hybrid element model is etablished.Using the thin plate bending element with varying thickness as an example, we compare various hybrid elements based on different generalized variational principles.     

非线性有限元的若干基本问题

本文介绍了非线性有限元中的若干基本问题。其中包括有关应变、应力和非线性平衡方程的一些基本概念,基于不同非线性广义变分原理的位移模式、杂交模式和拟协调模式几何非线性有限元及其在壳体屈曲问题中的应用等。      

Theoretical Formulation of Finite-Element Methods in Linear-Elastic Analysis of General Shells∗

A systematic classification of the variational functional whose stationarity conditions (Euler equations) can be used alternately to solve for the various unknowns in a boundary-value problem in linear-shell theory is made. The application of these alternate variational principles to a finite-element assembly of a shell and thus, the development of the properties of an individual discrete element are studied in detail. A classification of the finite-element methods, formulated from the variational principles by systematically relaxing the continuity requirements at the interelement boundaries of adjoining discrete elements is made.     

Generalized variational principles for boundary value problem of electromagnetic field in electrodynamics

An expression of the generalized principle of virtual work for the boundary value problem of the linear and anisotropic electromagnetic field is given. Using Chien＇s method, a pair of generalized variational principles （GVPs） are established, which directly leads to all four Maxwell＇s equations, two intensity-potential equations, two constitutive equations, and eight boundary conditions. A family of constrained variational principles is derived sequentially. As additional verifications, two degenerated forms are obtained, equivalent to two known variational principles. Two modified GVPs are given to provide the hybrid finite element models for the present problem.     

Variational principles of fluid full-filled elastic solids

IntroductionManypracticalproblemsinengineeringinvolveanalysisoffluidfijll-filledelasticsolids.Energyexplorationand"utilizationaretwoexamples.ThefieldequationsofBlot'sstaticalanddynamicaltheoryoffluidfijll-filledelasticsolidswereestablishedinRefs.[1,Zj.BecausetheitisdifficulttogetexactanswersInumericalmethodsareadopted,especiallythet'initCelementmethod.Theelementmethodbasedonvariationalprinciplesisappliedextensively.GhaboussiandWilsonderivedvariationalprinciplesonthebasisofBlot'sequationsan…     

薄板弯曲问题的P型杂交解析有限元方法

本文采用两套变量构造有限元试函数空间，在单元内部要求试函数精确满足平衡微分方程，在单元边界上对位移和转角分别用Ｐｅａｎｏ升阶函数插值，然后利用广义变分原理建立了一种薄板弯曲问题的Ｐ型杂交解析有限方法，与常规有限元法相比，该方法不心进行过细的网格剖分，通过增加单元插值多项式的阶数Ｐ来提高精度，此外，该方法还具有积分计算只需在单元边界上进行、单元钢度矩阵和载荷向量具有嵌入结构、协调程度可以自动控制等优     

扩展的Hellinger-Reissner原理及特殊层合杂交应力元

建立了一种非匀质材料新的、扩展的Hellinger-Reissner原理,发展了当一个单元域划分为不同材料特性子域、其元内应力场沿子域表面不连续、且位移场在子域表面也急剧变化时,一个非匀质有限元刚度列式便利方法。这种列式亦可用于对每层横向剪应变均独立处置的厚层板。基于此变分原理建立了新的具有一个无外力圆柱表面的层合杂交应力元,单元各层独立假定的应力场通过以自然坐标表示的非协调位移为权函数使齐次平衡方程变分满足的理性方法及严格满足给定圆柱面上无外力条件得到,位移场在元间及层间连续条件则分别通过Lagrange乘子进行了松弛。数值算例表明：这类新型元可有效地分析具有多类圆柱形槽孔的厚、中厚及薄层板自由孔边应力分布。     

P-version hybrid analytical finite element method for plate bending problems

Two sets of trial functions with different variables are constructed for the admissible space of the finite element analysis. The trial functions satisfy the equilibrium differential equation inside elements, while the deflections and rotations on the edges of the elements are approximated by the Peano hierarchical interpolation functions. Then, a generalized variational principle is applied to set up the p-version hybrid analytical finite element method for plate bending problems. The accuracy of finite element computation can be improved by increasing the order of the interpolation polynomials with fixed mesh. In the finite element formulation, to obtain the stiffness matrices and the load vectors, it is only necessary to perform quadrature over the edges of the elements. These matrices and vectors possess an embedding structure. The conformability between the elements can be controlled automatically.This work is supported by the Natural Science Foundation of China and the Aeronautical Science Foundation of China.     

基于开放式结构有限元系统 SiPESC.FEMS平板壳单元研发

基于开放式结构有限元系统SiPESC.FEMS的单元计算模块的设计模式,研发设计一种通用的平板壳单元计算框架。考虑板壳单元的组合关系和程序编制过程中的重用性及灵活性等特点,采用了软件设计中的构造器(Builder)模式实现不同的组合单元。本框架具有很好的通用性和可扩展性,为有限元程序研发提供了一个新的方式;同时,系统能够处理复杂荷载和边界条件,并可灵活实现不同类型单元的组合分析。本文利用此方法构造五种平板壳单元,通过数值算例分析对比讨论其性能,为选取合适的平板壳单元类型进行结构数值分析提供参考。     

Mixed compatible element and mixed hybrid incompatible element variational methods in dynamics of viscous barotropic fluids

ＭＩＸＥＤＣＯＭＰＡＴＩＢＬＥＥＬＥＭＥＮＴＡＮＤＭＩＸＥＤＨＹＢＲＩＤＩＮＣＯＭＰＡＴＩＢＬＥＥＬＥＭＥＮＴＶＡＲＩＡＴＩＯＮＡＬＭＥＴＨＯＤＳＩＮＤＹＮＡＭＩＣＳＯＦＶＩＳＣＯＵＳＢＡＲＯＴＲＯＰＩＣＦＬＵＩＤＳＳｈｅｎＸｉａｏ－ｍｉｎｇ（沈孝明...     

一类新型受任意载荷的杂交旋转壳单元

本文从弹性力学Reissner变分原理出发推导旋转壳曲线坐标系下内分，位移的二类变量广义变分原理，依据这个原理推导一类旋转壳坐标系中具有独立横向转角的受谐和外载荷下的杂交旋转壳单元，内力模式的选用使刚度矩阵的剪切部份在薄壳情况下能反映Kirchhoff假设，并使单元刚度矩阵满秩，从而保证单元无剪切自锁和零能模式，数例证明这类单元对中厚和薄旋转壳具有良好的通用性和较高的精度。     

Variational principles for hydrodynamic impact problems

We first establish the rigorous field equations of the two continuous stages before and after entering water. Then correspondently, we obtain the specific variational principles, bounded theorems, and boundary integral equations of the second stage problems. The existence of solutions are proved and the scheme of solving the solutions are provided. Finally, as a numerical example, the ship's wave resistence problem is used to demonstrate the specific application of the second stage problems and its accuracy. Then we provide a rigorous and sound theoretical basis of variational finite element method and boundary element method for calculating the accurately fundamental equations.     

Subregion generalized variational principles for elastic thick plates

Tn this paper, the subregion generalized variational principle for elastic thick plates is proposed. Its main points may be stated as follows:1. Each subregion may be assigned arbitrarily as a potential region or complementary region. The subregion variational principles of potential energy, complementary energy and mixed energy represent three special forms of this principle.2. The number of independent variational variables in each sub-region may be assigned arbitrarily. Any one of the subregions may be assigned as a one-variable-region, two-variable-region or three-variable-region.3. The conjunction conditions of displacements and stresses on each interline of neighbouring subregions may be relaxed. On the basis of this principle the finite element analysis of non-conforming elements for thick plates can be formulated.Different finite element models for thick plates can be obtained by different applications of this principle. In particular,the subregion mixed variational principle for thick plates may be applied to formulating the subregion mixed finite element method for thick plates.     

精化杂交压电固体单元的研究

基于力、电耦合问题的三类交量广义交分原理，提出了广义杂交压电单元列式。为了进一步改进单元的性能和保证单元能够通过分片检验，通过引入非协调模式、放松电学方程约束条件和单元间的弱连续性条件，建立了新的、修正的广义交分原理，在此基础上成功地引入了应力、应交的正交化插值模式，从而建立了精化杂交压电单元法，它继承了常规精化杂交单元的全部优点。文中所推导的八节点精化杂交压电固体单元列式完全避免了矩阵求逆运算，较广义杂交压电单元和杂交应力压电单元均显著提高了计算效率。数值算例表明，与同类型其他单元相比，该单元明显具有更好的对歪斜网格的适应性。     

非线性广义杂交元

选择新的三类变量建立了放松单元间连续性条件的用于非线性有限元分析的泛函，并由此建立了基于不协调模式的非线性广义杂交元方法。     

THE ANALYSIS OF SHALLOW SHELLS BASED ON MULTIVARIABLE WAVELET FINITE ELEMENT METHOD

Based on the generalized variational principle and B-spline wavelet on the interval (BSWI), the multivariable BSWI elements with two kinds of variables (TBSWI) for hyperboloidal shell and open cylindrical shell are constructed in this paper. Different from the traditional method, the present one treats the generalized displacement and stress as independent variables. So differentiation and integration are avoided in calculating generalized stress and thus the precision is improved. Furthermore, compared with commonly used Daubechies wavelet, BSWI has explicit expression and excellent approximation property and thus further guarantee satisfactory results. Finally, the effciency of the constructed multivariable shell elements is validated through several numerical examples.     

THE CONSTRUCTION OF WAVELET-BASED TRUNCATED CONICAL SHELL ELEMENT USING B-SPLINE WAVELET ON THE INTERVAL

Based on B-spline wavelet on the interval (BSWI), two classes of truncated conicalshell elements were constructed to solve axisymmetric problems, i.e. BSWI thin truncated conicalshell element and BSWI moderately thick truncated conical shell element with independent slope-deformation interpolation. In the construction of wavelet-based element, instead of traditionalpolynomial interpolation, the scaling functions of BSWI were employed to form the shape functionsthrough the constructed elemental transformation matrix,and then construct BSWI element viathe variational principle. Unlike the process of direct wavelets adding in the wavelet Galerkinmethod, the elemental displacement field represented by the coefficients of wavelets expansionwas transformed into edges and internal modes via the constructed transformation matrix. BSWIelement combines the accuracy of B-spline function approximation and various wavelet-basedelements for structural analysis. Some static and dynamic numerical examples of conical shellswere studied to demonstrate the present element with higher efficiency and precision than thetraditional element.     

The solid variational principles of the discrete from—The variational principles of the discrete analysis by the finite element method

This paper suggests a new solid variational principle of discrete form. Basing on the true case of the discrete analysis by the finite element method and considering the variable boundaries of the elements and the unknown functions of piecewise approximation, the unknown functions have various discontinuities at the interfaces between successive element.Thus, we have used mathematical technique of variable boundary with discontinuity of the unknown functions, based on the conditions that the first variation vanishes immediately, to establish the solid variation principles of discrete form. It generalizes the classical and non-classical variational principles. Successive equations that have to be satisfied by the unknown functions are the convergency necessary conditions for the finite elements method (including conforming and non-conforming). They expand that convergency necessary conditions of the compatibility conditions in the internal interfaces.     

Analysis of a strengthened weakly conical shell using a complete system of orthogonal functions on an arbitrary contour of its transverse cross-section

Thin-walled weakly conical and cylindrical shells with arbitrary open, simply or multiply closed contour of transverse cross-sections strengthened by longitudinal elements (such as stringers and longerons) are used in the design of wings, fuselages, and ship hulls. To avoid significant deformations of the contour, such structures are stiffened by transverse elements (such as ribs and frames). Various computational models and methods are used to analyze the stress-strain states of such compound structures. In particular, the ground stress-strain states in bending, transverse shear, and twisting of elongated structures are often analyzed with the use of the theory of thin-walled beams [1] based on the hypothesis of free (unconstrained) warping and bending of the contour of transverse cross-sections. In general, the computations with the contour warping and bending constraints caused by the variable load distribution, transverse stiffening elements, and the difference in the geometric and rigidity parameters of the shell cells are usually performed by the finite element method or the superelement (substructure) method [2, 3]. In several special cases (mainly for separate cells of cylindrical and weakly conical shells located between transverse stiffening elements, with the use of some additional simplifying assumptions), efficient variation methods for computations in displacements [4–8] and in stresses [9] were developed, so that they reduce the problem to a system of ordinary differential equations. In the one-and two-term approximations, these methods permit obtaining analytic solutions, which are convenient in practical computations. But for shells with multiply closed contours of transverse cross-sections and in the case of exact computations by using the Vlasov variational method [4], difficulties are encountered in choosing the functions representing the warping and bending of the contour of transverse cross-sections. In [10], in computations of a cylindrical shell with simply closed undeformed contour of the transverse section, warping was represented in the form of expansions in the eigenfunctions orthogonal on the contour, which were determined by the method of separation of variables from a special integro-differential equation. In [11], a second-order ordinary differential equation of Sturm-Liouville type was obtained; its solutions form a complete system of orthogonal functions with orthogonal derivatives on an arbitrary open simply or multiply closed contour of a membrane cylindrical shell stiffened by longitudinal elements. The analysis of such a shell with expansion of the displacements in these functions leads to ordinary differential equations that are not coupled with each other. In the present paper, by using the method of separation of variables, we obtain differential and the corresponding variational equations for numerically determining complete systems of eigenfunctions on an arbitrary contour of a discretely stiffened membrane weakly conical shell and a weakly conical shell with undeformed contour. The obtained systems of eigenfunctions are used to reduce the problem of deformation of shells of these two types to uncoupled differential equations, which can be solved exactly.