ON THE METHOD OF ORTHOGONALITY CONDITIONS FOR SOLVING THE PROBLEM OF LARGE DEFLECTION OF CIRCULAR PLATE

In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zangfor solving the problem of large deflection of a circular plate, and find that the method could be regarded as the method of strained parameters in the singular perturbation theory. In terms of the parameter representing the ratio of the center deflection to the thickness of the plate, we make the asymptotic expansions of the deflection, membrane stress and the parameter of load as in Ref. [1], and then give the orthogonality conditions (i.e. the solvability conditions) for the resulting equations, by which the stiffness characteristics of the plate could be determined. It is pointed out that with the solutions for the small deflection problem of the circular plate and the orthogonality conditions, we can derive the third order approximate relations between the parameter of load and the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases (i.e. under uniform load, under compound toad, with different boundary conditiors), we deduce the specific expressions and obtain the results in agreement with the previous ones given by Chien Wei-zang, Yeh kai-yuan and Hwang Chien in Refs. [1 - 4J.

Construction and analysis of the quadratic finite volume methods on tetrahedral meshes

We construct and analyze a family of quadratic finite volume method(FVM)schemes over tetrahedral meshes.In order to prove the stability and the error estimate,we propose the minimum V-angle condition on tetrahedral meshes,and the surface and volume orthogonal conditions on dual meshes.Through the technique of element analysis,the local stability is equivalent to a positive definiteness of a 9 × 9 element matrix,which is difficult to analyze directly or even numerically.With the help of the surface orthogonal condition and the congruent transformation,this element matrix is reduced into a block diagonal matrix,and then we carry out the stability result under the minimum V-angle condition.It is worth mentioning that the minimum V-angle condition of the tetrahedral case is very different from a simple extension of the minimum angle condition for triangular meshes,while it is also convenient to use in practice.Based on the stability,we prove the optimal H^(1) and L^(2) error estimates,respectively,where the orthogonal conditions play an important role in ensuring the optimal L^(2) convergence rate.Numerical experiments are presented to illustrate our theoretical results.

New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method

New superconvergent structures are proposed and analyzed for the finite volume element(FVE)method over tensorial meshes in general dimension d(for d≥2);we call these orthogonal superconvergent structures.In this framework,one has the freedom to choose the superconvergent points of tensorial k-order FVE schemes(for k≥3).This flexibility contrasts with the superconvergent points(such as Gauss points and Lobatto points)for current FE schemes and FVE schemes,which are fixed.The orthogonality condition and the modified M-decomposition(MMD)technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes.Numerical experiments are provided to validate our theoretical results.

On the Construction and Analysis of Finite Volume Element Schemes with Optimal L^(2) Convergence Rate

We provide a general construction method for a finite volume element(FVE)scheme with the optimal L^(2)convergence rate.The k-(k-1)-order orthogonal condition(generalized)is proved to be a sufficient and necessary condition for a k-order FVE scheme to have the optimal L^(2) convergence rate in 1D,in which the independent dual parameters constitute a(k-1)-dimension surface in the reasonable domain in k-dimension.In the analysis,the dual strategies in different primary elements are not necessarily to be the same,and they are allowed to be asymmetric in each primary element,which open up more possibilities of the FVE schemes to be applied to some complex problems,such as the convection-dominated problems.It worth mentioning that,the construction can be extended to the quadrilateral meshes in 2D.The stability and H^(1) estimate are proved for completeness.All the above results are demon-strated by numerical experiments.