THE ESTABLISHMENT OF BOUNDARY INTEGRAL EQUATIONS BY GENERALIZED FUNCTIONS

By the theory of generalized functions this paper introduces a specific generalizedfunction δθβ,by which,togehter with its various derivatives,the boundary integralequations and its arbitrary derivatives of any sufficiently smooth function can beestablished.These equations have no non-integral singlarities.For a problem defined bylinear partial differential operators,the partial differential equations of the problem canalways be converted into boundary integral equations so long as the relevant fundamentalsolutions exist.     

THE APPLICATION OF THE FINITE ELEMENT METHOD TO SOLVING BIOT’S CONSOLIDATION EQUATION

Biot’s theory of consolidation of saturated soil regards the con-solidation process as a coupling problem between stress of elas-tic body and flow of fluid existing in pores.It can moreprecisely reflect the mechanism of consolidation than Terzhi-gi’s theory.In this article,we obtain the general Biot’sfinite element equations of consolidation with classical varia-tional principles.The equations have clear physical meaningand have been applied to analysing the consolidation of Bajia-zui earth dam.The computational results are in accord withengineering practice.     

MORE GENERALIZED HYBRID VARIATIONAL PRINCIPLE AND CORRESPONDING FINITE ELEMENT MODEL

According to recent studies of the generalized variational principle by Professor ChienWeizang,the more generalized hybrid variational principle for finite element method isgiven,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,wecompare various hybrid elements based on different generalized variational principles.     

FUZZY ARITHMETIC AND SOLVING OF THE STATIC GOVERNING EQUATIONS OF FUZZY FINITE ELEMENT METHOD

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables , was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.     

THE APPLICATION OF GENERALIZED VARIATIONAL PRINCIPLE IN FINITE ELEMENT-SEMIANALYTICAL METHOD

The method developed in this paper is inspired by the siewpoint in ref. [1] that sufficient attention has not been paid to the value of the generalized variational principle in dealing with the houndary conditions in the finite element method. This method applies the generalized variational principle and chooses the series constituted by spline function multiplied by sinusoidal function and added by polynomial as the approximate deflection of plates and shells. By taking the deflection problem of thin plate, it shows that this method can solve the coupling problem in the finite element-semianalytical method. compared with the finite element method and finite stripe method, this method has much fewer unknown variables and higher precision. Hence, it proposer an effective way to solve this kind of engineering problems by minicomputer.     

THE EXACT INTEGRAL EQUATION OF HERTZ’S CONTACT PROBLEM

This paper presents the exact integral equation of Hertz‘s contact problem,which isobtained by taking into account the horizontal displacement of points in the contactedsurfaces due to pressure.     

THE SOLID VARIATIONAL PRINCIPLES OF THE DISCRETE FORM—THE VARIATIONAL PRINCIPLES OF THE DISCRETE ANALYSIS BY THE FINITE ELEMENT METHOD

This paper suggests a new solid variational prin-ciple of discrete form.Basing on the true case of thediscrete analysis by the finite element method and con-sidering the variable boundaries of the elements andthe unknown functions of piecewise approximation,theunknown functions have various discontinuities at theinterfaces between successive elements.Thus,we have used mathematical technique of vari-able boundary with discontinuity of the unknown func-tions,based on the conditions that the first varia-tion vanishes immediately,to establish the solid vari-ation principles of discrete form.It generalizes theclassical and non-classical variational principles.suc-cessive equations that have to be satisfied by the un-known functions are the convergency necessary conditionsfor the finite elements method(including conformingand non-conforming).They expand that convergency ne-cessary-conditions of the compatibility conditions inthe internal interfaces.     

A RECTANGULAR ELEMENT OF THIN PLATES BASED UPON THE GENERALIZED VARIATIONAL PRINCIPLES

Based on generalized variational principles,an element called MR-12 was constructed for the static and dynamic analysis of thin plates with orthogonal anisotropy.Numerical results showed that this incompatible clement converges very rapidly and has good accuracy.It was demonstrated that generalized variational principles are useful and effective in founding incompatible element.Moreover,element MR-12 is easy for implementation since it does not differ very much from the common rectangular element R-12 of thin plate.     

VARIATIONAL PRINCIPLES AND GENERALIZED VARIATIONAL PRINCIPLES FOR NONLINEAR ELASTICITY WITH FINITE DISPLACEMENT

In a previous paper (1979), the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain     

ON THE EMBEDDING AND COMPACT PROPERTIES OF FINITE ELEMENT SPACES

In this paper, the generalized Sobolev embedding theorem and the generalized Rellich-Kondrachov compact theorem for finite element spaces with multiple sets of functions are established. Specially, they are true for nonconforming, hybrid and quasi-conforming element spaces with certain conditions.     

Radiation boundary conditions for finite element solutions of generalized wave equations

On the basis of the dispersion relation of the generalized linear wave equation we derive a radiation boundary condition (RBC) that explicitly incorporates the physical parameters of the governing equation into the form of the boundary condition. Using finite element techniques we investigate the properties of the generalized RBC by examining forced and unforced solutions to the telegraph and Klein-Gordon equations in one dimension. The results show that within the limits of the physical parameters of the problem the generalized RBC is an improvement over the Sommerfeld RBC when the governing equation contains additional terms that influence the propagation. These gains are achieved without introducing any computational overhead. A two-dimensional example suggests that the 1D findings can generalize to higher dimensions.     

The establishment of boundary integral equations by generalized functions

By the theory of generalized functions this paper introduces a specific generalized function _p by which, together with its various derivatives, the boundary integral equations and its arbitrary derivatives of any sufficiently smooth function can be established. These equations have no non-integral singularities. For a problem defined by linear partial differential operators, the partial differential equations of the problem can always be converted into boundary integral equations so long as the relevant fundamental solutions exist.This paper is partial fulfilment of the first author's doctorial dissertation and the second author is the endviser.     

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.     

Crack approaching perpendicularly the boundary of a rectangular sheet: An integral equation and finite element solution

A series of crack problems are confronted combining the method of Singular-Integral Equation and the method of Finite-Element with the help of Schwarz's Alternating Method. In this way the capability of the Singular-Integral Equations Method to describe accurately singular fields is complemented by the ability of the Finite-Element Method to solve bodies with complicated boundaries. Applications are made in the case of a crack approaching perpendicularly a straight boundary of a finite sheet and in the case of an edge crack. The results, namely the values of stress intensity factors at the crack-tips, are satisfactory although few classical elements are used.     

On the variational boundary integral equations in elastodynamics with the use of conjugate functions

This paper proposes a Variational Boundary Integral Equation for time harmonic elasticity, using conjugate functions. A bilinear hermitian form for the variational formulation, as well as an a posteriori error indicator are proposed. The method does not involve hypersingular integrals in the finite part sense and preserves the symmetrical structures of equations.     

Criteria for finite element algorithm of generalized heat conduction equation

To eliminate oscillation and overbounding of finite element solutions of classical heat conduction equation, the author and Xiao have put forward two new concepts of monotonies and have derived and proved several criteria. This idea is borrowed here to deal with generalized heat conduction equation and finite element criteria for eliminating oscillation and overbounding are also presented. Some new and useful conclusions are drawn.     

The random variational principle and finite element method

In this paper, we introduced the random materials, geometrical shapes, force and displacement boundary condition directly into the functional variational formulations and developed a unified random variational principle and finite element method with the small parameter perturbation method. Numerical examples showed that the methods have the advantages of the simple and convenient program implementation, and are effective for the random mechanics problems.     

Analysis of linear elastic wave propagation in piping systems by a combination of the boundary integral equations method and the finite element method

The combined methodology of boundary integral equations and finite elements is formulated and applied to study the wave propagation phenomena in compound piping systems consisting of straight and curved pipe segments with compact elastic supports. This methodology replicates the concept of hierarchical boundary integral equations method proposed by L. I. Slepyan to model the time-harmonic wave propagation in wave guides, which have components of different dimensions. However, the formulation presented in this article is tuned to match the finite element format, and therefore, it employs the dynamical stiffness matrix to describe wave guide properties of all components of the assembled structure. This matrix may readily be derived from the boundary integral equations, and such a derivation is superior over the conventional derivation from the transfer matrix. The proposed methodology is verified in several examples and applied for analysis of periodicity effects in compound piping systems of several alternative layouts.     

The random variational principle in finite deformation of elasticity and finite element method

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Singular integral equations and boundary element method of cracks in thermally stressed planar solids

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