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PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Improvements in the Parametric Meshless Galerkin Method
H. Hosseini-Toudeshky and M. Musivand-Arzanfudi
Department of Aerospace Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
H. Hosseini-Toudeshky, M. Musivand-Arzanfudi, "Improvements in the Parametric Meshless Galerkin Method", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 180, 2006. doi:10.4203/ccp.83.180
Keywords: parametric, meshless, Galerkin method, parametric study, accuracy, computational time.
Meshless or meshfree techniques are alternative methods to eliminate the well-known drawbacks of the finite element methods due to the connectivity between the elements and nodes. Meshless methods besides many benefits, have shortcomings when compared to the finite element method (FEM). Computation of meshless shape functions and their derivatives is usually more complicated and time consuming than the FEM shape functions and is performed within the running time.
Recently, a new meshless method called the parametric meshless Galerkin method (PMGM) has appeared in the field of meshless methods . In this method, meshless shape functions are constructed in a parametric space. For this purpose the so called meshless parametric domains (MPDs) are created and stored in a library similar to the elements library of the commercial finite element softwares. In order to model the new problems, the shape functions are mapped from the parametric space to the physical space. Therefore the method is faster than the previously existed meshless methods, due to the shape function computational time saving. In addition to the parameters such as the size and shape of the influence domain, the weight functions, the basis functions, and the integration scheme, in the existing meshless methods, the accuracy and efficiency of the PMGM is highly dependent on two new parameters, the so called clearance and exceeding margin . The influence domain size of the extra-nodes  may also be adopted as a new parameter specific to the PMGM. The clearance parameter is the minimum distance between the nodes of the parametric domain from the edges of that. A non-zero clearance is necessary to avoid coincidence of the nodes in the final model. This parameter will also affect the final arrangement of the nodes in the problem domain. The exceeding margin parameter is the size of a strip outside of the parametric domain covered with the influence domains of the nodes of the parametric domain.
In this paper a parametric study is carried out to determine the best values for the important parameters of the PMGM. For this purpose two different problems in the two dimensional elasticity are analyzed using the PMGM. The exact solutions are available for both problems. The first problem is a cantilever beam which is subjected to shear traction at the free end. The second problem is a pressure-loaded half plane. Several configurations of the nodal arrangements have been considered for each of these two problems. The accuracy of displacements and stresses, the error norms for displacements and energy and the computational times of the defined problems are studied considering various values of the parameters of the clearance, exceeding margin, influence domain size, basis functions and Gaussian integration order, in order to determine the best values for these parameters.
It was seen that the accuracy of the results generally increases with increasing the size of the exceeding margin and the computational time saving decreases with increasing the size of the exceeding margin. It was suggested to choose the size of exceeding margin as 0.17 times of the edge length of the MPD, which may be a choice from both the accuracy and time saving viewpoints. The results show that the large values of influence domain of the extra-nodes decrease the saving of computational time achieved with the PMGM. It has been suggested to choose the value of the influence domain sizes of extra-nodes as half of the influence domain sizes of the neighbour nodes which are mapped from the MPDs. The obtained results show that this choice gives more stable results and leads to a more computational time saving. It has been also observed that adding the extra-nodes to the regions between adjacent approximation subdomains generally has not significant effects on the results. Based on the results of the analyses, the value of the clearance was suggested to be 0.375 times of the nodal spacing of the MPD. For the beam problem, the quadratic basis gives more accurate results than the linear basis, while for the half plane problem there are not significant differences between the results obtained by linear and quadratic basis. This is due to the fact that the quadratic basis can reproduce the exact solution of the beam problem. It is noticed that using the quadratic basis needs more computational time than the linear basis, similar to the other meshless methods. Therefore using the linear basis is more favourable except for the problems which their solutions are known to be explained by quadratic functions. The parametric study on the Gaussian integration order shows that 4x4 Gaussian points in each integration cell is a reasonable choice for both linear and quadratic basis and both beam and half plane problems, when one Gaussian integration cell exists per each node of the MPD. The stress diagrams which were computed using the suggested parameters were in excellent agreement with the exact solutions.
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