COMPARATIVE STUDY OF THE WEAK-FORM COLLOCATION MESHLESS FORMULATION AND OTHER MESHLESS METHODS
DOI:
https://doi.org/10.26512/ripe.v2i6.21472Keywords:
Local Meshless. Generalized-strain. Weak-form collocation. Local work theorem. Comparative study.Abstract
This paper is concerned with the numerical comparison of the weak-form collocation, a new local meshless method, and other meshless methods, for the solution of two-dimensional problems in linear elasticity. Four methods are compared, namely, the Generalized-Strain Mesh-free (GSMF) formulation, the Rigid-body Displacement Mesh-free (RBDMF) formulation, the Element-free Galerkin (EFG) and the Meshless Local Petrov-Galerkin Finite Volume Method (MLPG FVM). While the RBDMF, EFG and MLPG FVM rely on integration and quadrature process to obtain the stiffness matrix, the GSMF is completely integration free, working as a weighted-residual weak-form collocation. This weak-form collocation readily overcomes the well-known difficulties of the strong-form collocation, such as low accuracy and instability of the solution. A numerical example was analyzed with these methods, in order to assess the accuracy and the computational effort. The results obtained are in agreement with those of the available analytical solution. The numerical results show that the GSMF, when compared to the other methods, is superior not only regarding the computational efficiency, but also regarding the accuracy.
Downloads
References
Atluri, S.N. and S. Shen (2002). “The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple and Less-costly Alternative to the Finite Element and Boundary Element Methods”.In: CMES: Computer Modeling in Engineering and Sciences 3.1, pp. 11”“51.
Atluri, S.N. and T. Zhu (1998). “A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics”. In: Computational Mechanics 22.2, pp. 117”“127.
Atluri, S.N., Z.D. Han, and A.M. Rajendran (2004). “A New Implementation of the Meshless Finite Volume Method Through the MLPG Mixed Approach”. In: CMES: Computer Modeling in Engineering and Sciences 6, pp. 491”“513.
Belytschko, T., Y. Y. Lu, and L. Gu (1994). “Element-free Galerkin methods”. In: International Journal for Numerical Methods in Engineering 37.2, pp. 229”“256. ISSN: 1097-0207.
Brebbia, C.A. and H. Tottenham, eds. (1985). Variational Basis of Approximate Models in Continuum Mechanics. The II International Conference on Variational Methods in Engineering. Berlin: Southampton and Springer Verlag.
Fichera, G (2006). Linear Elliptic Differential Systems and Eigenvalue Problems. Springer.
Finalyson, B.A. (1972). The Method of Weighted Residuals and Variational Principles. Vol. 87.
Academic Press, p. 472.
Fredholm, I. (1906). “Solution d’un problme fondamental de la thorie de l’lasticit”. In: Arkiv fr Matematik Astronomi och Fysk 2 28.1, pp. 1”“8.
Gelfand, I.M. and G.E. Shilov (1964). Generalized Functions. Vol. 1. Academic Press.
Liu, G.R. and Y.T. Gu (2001). “A Local Point Interpolation Method for Stress Analysis of Two-Dimensional Solids”. In: Structural Engineering and Mechanics 11.2, 221”“236.
Liu, G.R. et al. (2002). “Point Interpolation Method Based on Local Residual Formulation Using Radial Basis Functions”. In: Structural Engineering and Mechanics 14, 713”“732.
Liu, W.K., S. Jun, and Y.F. Zhang (1995). “Reproducing Kernel Particle Methods”. In: International Journal for Numerical Methods in Engineering 20, 1081”“1106.
Nayroles, B., G. Touzot, and P. Villon (1992). “Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements”. In: Computational Mechanics 10, 307”“318.
Oliveira, T. and A. Portela (2016). “Weak-Form Collocation ”“ a Local Meshless Method in Linear Elasticity”. Engineering Analysis with Boundary Elements. Submitted.
Sokolnikoff, I. S. (1956). Mathematical theory of elasticity. Vol. 83. McGraw-Hill New York.
Zhu, T., J. Zhang, and S.N. Atluri (1998). “A Local Boundary Integral Equation (LBIE) Method in Computational Mechanics and a Meshless Discretization Approach”. In: Computational Mechanics 21, 223”“235.
Downloads
Published
How to Cite
Issue
Section
License
Given the public access policy of the journal, the use of the published texts is free, with the obligation of recognizing the original authorship and the first publication in this journal. The authors of the published contributions are entirely and exclusively responsible for their contents.
1. The authors authorize the publication of the article in this journal.
2. The authors guarantee that the contribution is original, and take full responsibility for its content in case of impugnation by third parties.
3. The authors guarantee that the contribution is not under evaluation in another journal.
4. The authors keep the copyright and convey to the journal the right of first publication, the work being licensed under a Creative Commons Attribution License-BY.
5. The authors are allowed and stimulated to publicize and distribute their work on-line after the publication in the journal.
6. The authors of the approved works authorize the journal to distribute their content, after publication, for reproduction in content indexes, virtual libraries and similars.
7. The editors reserve the right to make adjustments to the text and to adequate the article to the editorial rules of the journal.
