The paper presents a fully meshless procedure fo solving partial differential equations. The approach termed generically the ‘finite point method’ is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals. Some examples showing the accuracy of the method for solution of adjoint and non-self adjoint equations typical of convective-diffusive transport and also to the analysis of compressible fluid mechanics problem are presented.

A finite point method in computational mechanics. applications to convective transport and fluid flow

On some convergence results for FDM with irregular mesh

The article traces the important steps of the development of the finite element method from its origins in aircraft structural engineering to the present day, where it provides the essential tool for solution of a great variety of problems in engineering and physics. The emphasis and the choice of the “landmarks” stresses the aspects which are general and essentially of mathematical nature applicable to a wide range of situations. For this reason no mention is made of perhaps equally important developments to new application fields such as metal forming, electromagnetics, geomechanics etc.

http://www.me.metu.edu.tr/courses/me582/files/Zienkiewicz_History_of_FEM.pdf

Origins, milestones and directions of the finite element method— A personal view

Presented is a concept of combination and formal unification of the Finite Element Method and Generalized Finite Difference Method at arbitrary irregular grids. Higher order FD approximation and a formulation on differential manifold may be also applied. Symbolic operations were used in order to provide a fully automatic approach in both the FE and FD methods. Results of numerical tests are discussed.

A Unified Approach to the FE and Generalized Variational FD Methods in Nonlinear Mechanics, Concepts and Numerical Approach

This paper presents some recent advances on the numerical solution of the classical Germain-Lagrange equation for plate bending of thin elastic plates. A meshless strategy using the Generalized Finite Difference Method (GFDM) is proposed upon substitution of the original fourth-order differential equation by a system composed of two second-order partial differential equations. Mixed boundary conditions, variable nodal density and curved contours are some of the explored aspects. Simulations using very dense clouds and parallel processing scheme for efficient neighbor selection are also presented. Numerical experiments are performed for arbitrary plates and compared with analytical and Finite Element Method solutions. Finally, an overview of the procedure is presented, including a discussion of some future development.

/pdf/reduced-order-strategy-for-meshless-solution-of-plate-3alt835zon.pdf

Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method

: A new energy based finite difference analytical technique is introduced. The method incorporates certain energy concepts and the ability to use arbitrary, irregular meshes within the framework of the Finite Difference Method. This formulation reduces any governing partial differential equations to a set of difference equations containing partial derivatives up to and including the second order. Further certain strong simularities with the popular Finite Element Method are shown and the ability to solve problems with irregular boundaries is discussed. To demonstrate the Finite Difference Energy Method several plate bending problems are solved. (Author)

Finite difference energy techniques for arbitrary meshes applied to linear plate problems

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Finite difference energy techniques for arbitrary meshes applied to linear plate problems