This work is devoted to some recent developments in the Higher Order Approximation introduced to the Meshless Finite Difference Method (MFDM), and its application to the solution of boundary value problems in mechanics. In the MFDM, approximation of the sought function is described in terms of nodes rather than by means of any imposed structure like elements, regular meshes etc. Therefore, the MFDM, using arbitrarily irregular clouds of nodes using the Moving Weighted Least Squares (MWLS) approximation falls into the category of the Meshless Methods (MM). The MFDM, dating to early seventies, is one of the oldest and possibly the most developed one. In this paper considered are some techniques which lead to improvement of the MFDM solution�s quality. The main objective of this paper is the presentation and overview of new ideas and the development of the Higher Order solution approach in the MFDM provided by correction terms, preceded by a brief information about the current state-of-the art of this method. The main concept of the Higher Order Approximation (HOA) used here, is based on consideration of additional terms in the local Taylor expansion of the sought function. It shall be demonstrated that such a move may essentially improve, in many ways, efficiency and solution quality of the Higher Order MFDM. The Higher Order correction terms may be applied in many aspects of the MFDM solution approach. Among them one may distinguish the a-posteriori error estimation as well as adaptive solution process with multigrid strategy. Moreover, in the present work considered are: computational implementation of the Higher Order MFDM algorithms, examination of the above mentioned aspects using 1D and 2D benchmark tests, as well as an application of the Higher Order MFDM solution approach to selected boundary value problems in mechanics.

Meshless Finite Difference Method with Higher Order Approximation—Applications in Mechanics

A novel gradient smoothing method (GSM) is proposed in this paper, in which a gradient smoothing together with a directional derivative technique is adopted to develop the first- and second-order derivative approximations for a node of interest by systematically computing weights for a set of field nodes surrounding. A simple collocation procedure is then applied to the governing strong-from of system equations at each node scattered in the problem domain using the approximated derivatives. In contrast with the conventional finite difference and generalized finite difference methods with topological restrictions, the GSM can be easily applied to arbitrarily irregular meshes for complex geometry. Several numerical examples are presented to demonstrate the computational accuracy and stability of the GSM for solid mechanics problems with regular and irregular nodes. The GSM is examined in detail by comparison with other established numerical approaches such as the finite element method, producing convincing results.

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A gradient smoothing method (GSM) with directional correction for solid mechanics problems

https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5948

A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

Meshless Finite Difference Method (MFDM) is nowadays a powerful engineering tool for numerical analysis of boundary value problems. Nowadays, its computational capabilities are not fully used mainly due to the lack of suitable commercial software. This paper briefly presents current state-of-the-art of the MFDM solution approach as well as deals with the selected computational aspects of the MFDM. A set of Matlab functions written by the author is attached to the paper. Techniques for generation of nodes, MFD stars, formulas, equations as well as local approximation technique and numerical integration schemes are discussed there.

/pdf/selected-computational-aspects-of-the-meshless-finite-35qj7ykf1s.pdf

Selected computational aspects of the meshless finite difference method

Analysis of three-dimensional heat conduction in functionally graded materials by using a hybrid numerical method

Meshless methods are the subject of increasing interest nowadays. As such considered is here the adaptive finite difference method generalized for arbitrary irregular grids (GFDM). This is an integrated approach including original concepts of a’posteriori error analysis, solution smoothing mesh generation and modification as well as special emphasis posed on multigrid solution approach. Although the GFDM itself has been a well established method for years [15], its fully adaptive formulation has only recently been proposed and outlined [17,18,20]. The method is designed as a general and powerful tool of analysis of large and very large discrete boundary-value problems. A fully adaptive GFDM multigrid solution approach is outlined here. Presented are new concepts of prolongation, restriction and solution procedure.

P. Lezanski

Papers

Multigrid Approach to Adaptive Analysis of B. V. Problems by The Meshless GFDM