A local meshless analysis of dynamics problems / Uma análise local desordenada dos problemas dinâmicos

Flávio dos Ramos de Sousa Mendonça, Wilber Humberto Vélez Gómez, Artur Antônio de Almeida Portela

Abstract


This paper is concerned with new formulations of local meshfree numerical method, for the solution of dynamic problems in linear elasticity, Integrated Local Mesh Free (ILMF) method. The key attribute of local numerical methods is the use of a modeling paradigm based on a node-by-node calculation, to generate the rows of the global system of equations of the body discretization. In the local domain, assigned to each node of a discretization, the work theorem is kinematically formulated, leading thus to an equation of mechanical equilibrium of the local node, that is used by local meshfree method as the starting point of the formulation. The main feature of this paper is the use of a linearly integrated local form of the work theorem. The linear reduced integration plays a key role in the behavior of local numerical methods, since it implies a reduction of the nodal stiffness which, in turn, leads to an increase of the solution accuracy. As a consequence, the derived meshfree and finite element numerical methods become fast and accurate, which is a feature of paramount importance, as far as computational efficiency of numerical methods is concerned. The cantilever beam was analyzed with this technique, in order to assess the accuracy and efficiency of the new local numerical method for dynamic problems with regular and irregular nodal configuration. The results obtained in this work are in perfect agreement with Mesh-Free Local Petrov-Galerkin (MLPG) and the Finite Element Method (FEM) solutions.


Keywords


Local Meshfree numerical method, dynamic problems, Moving Least Squares (MLS), Integrated Local Mesh Free (ILMF), Mesh-Free Local Petrov-Galerkin (MLPG).

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References


Daxini, S. D. and Prajapati, J. M. A review on recent contribution of the meshfree methods to structure and fracture mechanics applications, the scientific word journal, 13 pages, 2014.

Finalyson, B. A. The Method of Weighted Residuals and Vibrational Principles. Academic Press, 1972.

Chen, J. S., Hillman, M. and Chi, S. W. Meshfree Methods: Progress made after 20 years, Journal of Engineering Mechanics 143, 4, 2017.

Nayroles, B., Touzot, G. and Villon, P. Generalized the Finite Element Method: Diffuse Approximation and Diffuse Elements, Computational Mechanics 10, 307–318, 1992.

Liu, W. K., Jun, S., and Zhang, Y.F. Reproducing Kernel Particle Methods, International Journal for Numerical methods in Engineering 20, 1081–1106, 2007.

Belytshko, T., Lu, Y. Y. and Gu, L. Element – free Galerkin methods, International Journal for Numerical Methods in Engineering 37(2), 229–256, 1994.

Atluri, S. N. and Zhu, T. A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics 22(2), 117–127, 1998.

Atluri, S. N. and Shen, S. The Meshless Local Petrov – Galerkin (MLPG) Method: A simple and Less-costly Alternative to the Finite Element and Boundary Element Methods, CMES: Computer Modeling in Engineering and Sciences 3(1), 11-51. 2002.

Zhu, T., Zhang, J. and Atluri, S. N. A Local Boundary Integral Equation (LBIE) Method in Computational Mechanics and a Meshless Discretization Approach, Computational Mechanics 21, 223–235, 1998.

Liu, G. R. and Gu, Y. T. A Local Point Interpolation Method for Stress Analysis of Two-Dimensional Solids, Structural Engineering and Mechanics 11(2), 221–236, 2001.

Atluri, S. N. and Han, Z. D. and Rajendran, A. M. A New Implementation of the Meshless Finite Volume Method Through the MLPG Mixed Approach, CMES: Computer Modeling in Engineering and Sciences 6, 491-513, 2004.

Oliveira, T. and Portela, A. Weak – Form Collocation – a Local Meshless Method in Linear Elasticity. Engineering Analysis with Boundary Elements 73, 144 – 160, 2016.

Kirchhoff, G. Uber das Gleichgewicht und die Bewegung einer uendlich diinnen elastischen Stabes. J. Reine Angew. Math. v. 56, p.p. 285 – 313, 1859.

Brebbia C. A. and Tottenham, H. Variational Basis of Aproximate Models in Continuum Mechanics. Southampton and Springer Verlag, 1985.

Atluri, S. N. and Zhu, T. New Concepts in Meshless Methods, International Journal for Numerical Methods in Engineering 6, 537-556, 2000.

LIU, G. R. e GU, Y. T. An introduction to meshfree methods and their programming. Springer Science & Business Media, 2005.




DOI: https://doi.org/10.34117/bjdv7n10-134

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