Meshless Local Petrov-galerkin Euler-bernoulli Beam Problems: a Radial Basis Function Approach

A radial basis function implementation of the meshless local Petrov-Galerkin (MLPG) method is presented to study Euler-Bernoulli beam problems. Radial basis functions, rather than generalized moving least squares (GMLS) interpolations, are used to develop the trial functions. This choice yields a computationally simpler method as fewer matrix inversions and multiplications are required than when GMLS interpolations are used. Test functions are chosen as simple weight functions as in the conventional MLPG method. Compactly and noncompactly supported radial basis functions are considered. The non-compactly supported cubic radial basis function is found to perform very well. Results obtained from the radial basis MLPG method are comparable to those obtained using the conventional MLPG method for mixed boundary value problems and problems with discontinuous loading conditions. Introduction Meshless methods are developed to overcome some of the disadvantages of the finite element method (FEM) such as discontinuous secondary variables across inter-element boundaries and the need for remeshing in large deformation problems. Recent literature shows extensive research on meshless methods and, in particular, the meshless local PetrovGalerkin (MLPG) method. The majority of literature published to date on the MLPG method presents variations of the method for C problems. 6 However, a comparatively limited amount of work 4, 7-10 is reported on the more complicated C problems. Atluri et al. 4 present an analysis of thin beam problems using a Galerkin implementation of the MLPG method. In reference 4, a generalized moving least squares (GMLS) approximation is used to construct the trial functions, and the test functions are chosen from the same space. In references 11-14, a meshless PetrovGalerkin implementation of the MLPG method is presented; the GMLS approximation is used to construct the trial functions, and the test functions are chosen from a different space. Closer scrutiny of these formulations shows that a large number of calculations are required to compute the first and second order derivatives of the moving least squares (MLS) trial functions. Hence, a computationally efficient alternative to the MLS trial functions is preferred. This paper demonstrates the use of radial basis interpolation functions in the meshless local PetrovGalerkin formulation for beam problems. The radial basis functions are simple, and the evaluation of the derivatives is simpler than for the traditional MLS approximations. In the present radial basis MLPG (RPG) formulation, simple weight functions are chosen as test functions, and Gaussian quadrature is used to integrate the weak form. The effectiveness of the RPG method is evaluated by applying the formulation to a variety of patch test and mixed boundary value problems. The outline of the paper is as follows: First, the moving least squares interpolation used in the conventional MLPG method is discussed as motivation for finding a more computationally efficient alternative. Next, an overview of radial basis functions (RBF) for C problems is presented; the shape functions obtained from radial basis interpolation are derived, and the shape functions obtained when polynomial basis functions are included in the interpolation are derived. The development of these radial basis shape functions is then expanded and repeated for beam problems. The system of algebraic equations developed from the local weak form of the governing differential equation and the chosen trial and test functions is presented. Patch test problems are used to validate the RPG method for different choices of radial basis function. Then, the RPG method is applied to mixed boundary value problems. Finally, the method is applied to a problem with discontinuous loading conditions. Interpolation Schemes In this section, the moving least squares interpolation scheme used in the conventional MLPG https://ntrs.nasa.gov/search.jsp?R=20040085788 2017-12-05T02:30:09+00:00Z * Structures and Materials Competency, Senior Technologist, Fellow AIAA † Lockheed Martin Space Operations ‡ Analytical and Computational Methods Branch, Member AIAA This material is a declared work of the U.S. Government and is not subject to copyright protection in the United States. American Institute of Aeronautics and Astronautics 1 method is discussed first. Then, two interpolation schemes involving radial basis functions (RBF) are presented. In the first scheme, radial basis functions alone are used to construct the shape functions. The second scheme is a hybrid that uses both radial basis functions and polynomial basis functions to construct the shape functions. The Moving Least Squares Interpolation A moving least squares (MLS) interpolation is a scheme that passes a smooth function through an assumed set of fictitious nodal values. The interpolation is performed such that the least squares error between the function and the nodal values is a minimum. 2 A schematic of the MLS interpolation is presented in Figure 1. moving least squares fit