A variational principle of perturbed motion on viscoelastic thin plates with applications

In this paper, in the light of the Boltzmann superposition principle in linear viscoelasticity, a mathematical model of perturbed motion on viscoelastic thin plates is established. The corresponding variational principle is obtained in a convolution bilinear form. For application the problems of free vibration, forced vibration and stability of a viscoelastic simply-supported rectangular thin plate are considered. The results show that numerical solutions agree well with analytical solutions. The project was supported by the National Natural Science Foundation of China (No. 19772027) and the Shanghai Municipal Development Foundation of Science and Technology (No. 98JC14032).     

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.     

Study on interface failure of shape memory alloy （SMA） reinforced smart structure with damages

Shape memory alloy (SMA) reinforced smart structure can be used to make structural shape and strength self-adapted and structural damage self-restrained. Although SMA smart structures without damages were extensively studied, researches on SMA smart structures with damages have rarely been reported thus far. In this paper, thermo-mechanical behaviors of SMA fiber reinforced smart structures with damages are analyzed through a shear lag model and the variational principle. Mathematical expressions of the meso-displacement field and the stress-strain field of a typical element with damages are obtained, and a failure criterion for interface failure between SMA fibers and matrix is established, which is applied to an example. Results presented herein may provide a theoretical foundation for further studies on integrity of SMA smart structures.The project supported by the National Natural Science Foundation of China (10072026, 50135030) and Aeronautical Science Foundation of China (01G52041)The English text was polished by Keren Wang.     

Solving frictional contact problems by two aggregate-function-based algorithms

Three dimensional frictional contact problems are formulated as linear complementarity problems based on the parametric variational principle. Two aggregate-function-based algorithms for solving complementarity problems are proposed. One is called the self-adjusting interior point algorithm, the other is called the aggregate function smoothing algorithm. Numerical experiment shows the efficiency of the proposed two algorithms. The project supported by the National Natural Science Foundation of China(10225212, 50178016, 10302007), the National Key Basic Research Special Foundation and the Ministry of Education of China The English text was polished by Ron Marshall.     

Non-local continuum model and its numerical simulation for localization problem

A non-local continuum model for strain-softening simply taking plastic strain or damage variable as a non-local variable is derived by using the additive decomposition principle of finite deformation gradient. At the same time, variational equations, their finite element formulations and numerical convoluted integration algorithm of the model in current configuration usually called co-moving coordinate system are given. Stability and convergence of the model are proven by means of the weak convergence theorem of general function and the convoluted integration theory. Mathematical and physical properties of the characteristic size for material or structure are accounted for within the context of a statistical weighted or kernel function, and way is investigated. Numerical simulation shows that this model is suitable for to analyzing deformation localization problems. The project supported by the National Natural Science Foundation of China (No. 19632030).     

A combined parametric quadratic programming and precise integration method based dynamic analysis of elastic-plastic hardening/softening problems

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems. The project supported by the National Key Basic Research Special Foundation (G1999032805), the National Natural Science Foundation of China (19872016, 50178016, 19832010) and the Foundation for University Key Teacher by the Ministry of Education of China     

New axisymmetric solid elements based on an extended Hellinger-Reissner principle

4-node incompatible axisymmetric hybrid stress elements are developed by a modified rational approach based on an extended Hellinger-Reissner principle. The stress fields are derived by using the constraint conditions that the equilibrium equations of higher order stress terms are imposed in a variational sense through the additional incompatible displacements. The resulting elements enjoy good performance in bending problems and at the nearly incompressible limit. Several examples are given to demonstrate the performance of the new elements. The project supported by the National Natural Science Foundation of China     

A finite element model for numerical simulation of thermo-mechanical frictional contact problems

Two kinds of variational principles for numerical simulation of heat transfer and contact analysis are respectively presented. A finite element model for numerical simulation of the thermal contact problems is developed with a pressure dependent heat transfer constitutive model across the contact surface. The numerical algorithm for the finite element analysis of the thermomechanical contact problems is thus developed. Numerical examples are computed and the results demonstrate the validity of the model and algorithm developed. The project supported by the National Key Basic Research Special Foundation (G1999032805), the National Natural Science Foundation of China (50178016, 10225212) and the Foundation for University Key Teacher by the Ministry of Education of China     

参变量变分原理的提出、发展与应用

参变量变分原理及其参数二次规划算法是由钟万勰院士1985年针对弹性接触边界非线性问题首次提出来的,经过将近40年的不断发展,目前参变量变分原理已经成功应用于各个领域,其中包括弹塑性分析、接触问题、润滑力学、岩土力学、变刚度杆系结构、先进材料性能分析、材料的蠕变与损伤、柔性结构力学和LQ最优控制等各个工程领域。本文首先回顾了参变量变分原理的起源,介绍了参变量变分原理的基本概念,然后以弹塑性分析问题为例,阐明建立参变量变分原理的理论模型以及实现数值参数二次规划求解原理,最后详细回顾了参变量变分原理的基本理论与相应数值算法在各个领域的发展及其工程应用,展示了参变量变分原理在求解各类非线性问题的特色与优势。     

A penalty-hybrid finite element analysis of stokes flow

A type of penalty-hybrid variational principle is suggested for the analysis of Stokesian flow. On such a basis, a finite element model is formulated featuring, among others, a priori satisfaction of the deviatoric stress and hydrostatic pressure on linear momentum balance equations. Also in the present scheme the hydrostatic pressure is successfully eliminated at the element level, leaving only nodal velocities as solution unknowns. A series of 4-node and 8-node quadrilateral elements are derived and examined. Numerical examples demonstrating their characteristic behaviors are also included.The project Supported by National Natural Science Foundation of China     

On the Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings in Banach spaces

The purpose of this paper is to introduce and study the existence of solutions and convergence of Mann and Ishikawa iterative processes for a class of variational inclusions with accretive type mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results by Chang, Ding, Hassouni, Kazmi, Siddiqi, Zeng et al. Foundation item: the National Natural Science Foundation of China (19771058)     

A divided region variational principle ofA, ϕ-Ω method for 3-D eddy current problems

In this paper, a divided region variational principle to solve 3-D eddy current problems was given. It adopts the magnetic vector potential A and the electric scalar potential ϕ in the eddy current regions and the source regions, and the magnetic scalar potential Ω in the non-conducting regions (air gap). Using variation of the functional, all governing equations in various regions, the natural boundary conditions and the interface continuity conditions study which satisfy electromagnetic continuity are obtained. Project supported by the National Natural Science Foundation of China (59375197)     

Energy principles in theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], various energy principles in theory of elastic materials with voids can be established systematically. In this paper, an important integral relation is given, which can be considered essentially as the generalized pr. inciple of virtual work. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem of work in theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field generalized variational principles, and the principle of minimum potential and complementary energies. Furthermore, with this appro ach, the intrinsic relationship among various principles can be explained clearly. The project supported by the National Natural Science Foundation of China     

A study on non-steady flow of upper-convected maxwell fluid in tube

The transient response of an upper-convected Maxwell fluid flow in a circular tube is analysed by variational approach of Kantorovich and the method of finite difference. The solution of the variational method is in agreement with the numerical results by the difference schemes. The results show that the method of Kantorovich is suitable for the study of non-steady flow of non-Newtonian fluids and the effect of elasticity of the fluid has an influence on the non-steady flow. project supported by National Natural Science Foundation of China     

A variational analytical approach to time dependent flow of oldroyd B fluid in circular tube

In the present investigation the time dependent flow of an Oldroyd fluid B in a horizontal cylindrical pipe is stuided by the variational analytical approach developed by author. The time dependent problem is mathematically reduced to a partial differential equation of third order. Using the improved variational approach due to Kantorovich the partial differential equation can be reduced to a system of ordinary differential equations for different approximations. The ordinary differential equations are solved by the method of the Laplace transform which is led to an analytical form of the solutions. Project supported by TWAS and Chinese Academy of Sciences and the National Science Foundation of China     

A, φ-Ω METHOD FOR 3-D EDDY CURRENT ANALYSIS

After the field equations and the snonumuoo conditions between the interfaces for 3D eddy current problems Under various gauges were discussed, it was pointed cut in this paper that using the magnetic vector potential A. the electric scalar potential φ and Coulomb gauge ∇·A = 0 in eddy current regions and using the magntetic scalar potential Ω in the non-conducting regions are more suitable. All field equations, the boundary conditions, the interface continuity conditions and the corresponding variational principle of this method are also given.     

Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm

In this paper, the shallow water problem is discussed. By treating the incompressible condition as the constraint, a constrained Hamilton variational principle is presented for the shallow water problem. Based on the constrained Hamilton variational principle, a shallow water equation based on displacement and pressure (SWE-DP) is developed. A hybrid numerical method combining the finite element method for spatial discretization and the Zu-class method for time integration is created for the SWEDP. The correctness of the proposed SWE-DP is verified by numerical comparisons with two existing shallow water equations (SWEs). The effectiveness of the hybrid numerical method proposed for the SWE-DP is also verified by numerical experiments. Moreover, the numerical experiments demonstrate that the Zu-class method shows excellent performance with respect to simulating the long time evolution of the shallow water.     

Discrete mechanics and the Finite Element Method

Summary We explore applications of the Finite Element Method (FEM) to both Veselov and Lee discrete mechanics in this paper. Based on the FEM, disretizations of continuous Lagrangians are developed and corresponding integrators are obtained. Error estimates for variational integrators are also given. This work is supported by the National Natural Science Foundation of China (grant Nos. 90103004, 10171096) and the National Key Project for Basic Research of China (G1998030601).     

A generalized variational principle of composite shallow shells and its application to the folded shell

In this paper, a generalized variational principle of elastodynamics in composite shallow shells with edge beams is presented, and its equivalence to corresponding basic equations, ridge conditions and boundary conditions is proved. Then this variational principle is applied to the folded shell structure. By means of double series, the approximate analytical solutions for statics and dynamics under common boundary conditions are obtained. The comparison of our results with FEM computations and experiments shows the analytical solutions have good convergence and their accuracy is quite satisfactory.     

NONLINEAR VIBRATION FOR MODERATE THICKNESS RECTANGULAR CRACKED PLATES INCLUDING COUPLED EFFECT OF ELASTIC FOUNDATION

IntroductionThe moderate thickness plates on elastic foundation are a kind of important structure instructural engineering. The mechanic characters of the plates on elastic foundation withdifferent boundary conditions have been received considerable atten…