TY - JOUR

T1 - The Meshless Local Petrov-Galerkin (MLPG) method

T2 - A simple & less-costly alternative to the finite element and boundary element methods

AU - Atluri, Satya N.

AU - Shen, Shengping

PY - 2001

Y1 - 2001

N2 - A comparison study of the efficiency and accuracy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are presented in this paper. In all these six MLPG methods, absolutely no meshes are needed either for the interpolation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells. Because complicated shape functions for the trial function are inevitable at the present stage, in order to develop a fast and robust meshless method, we explore ways to avoid the use of a domain integral in the weak-form, by choosing an appropriate test function. The MLPG5 method (wherein the local, nodal-based test function, over a local sub-domain Ωs (or Ωte) centered at a node, is the Heaviside step function) avoids the need for both a domain integral in the attendant symmetric weak-form as well as a singular integral. Convergence studies in the numerical examples show that all of the MLPG methods possess excellent rates of convergence, for both the unknown variables and their derivatives. An analysis of computational costs shows that the MLPG5 method is less expensive, both in computational costs as well as definitely in human-labor costs, than the FEM, or BEM. Thus, due to its speed, accuracy and robustness, the MLPG5 method may be expected to replace the FEM, in the near future.

AB - A comparison study of the efficiency and accuracy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are presented in this paper. In all these six MLPG methods, absolutely no meshes are needed either for the interpolation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells. Because complicated shape functions for the trial function are inevitable at the present stage, in order to develop a fast and robust meshless method, we explore ways to avoid the use of a domain integral in the weak-form, by choosing an appropriate test function. The MLPG5 method (wherein the local, nodal-based test function, over a local sub-domain Ωs (or Ωte) centered at a node, is the Heaviside step function) avoids the need for both a domain integral in the attendant symmetric weak-form as well as a singular integral. Convergence studies in the numerical examples show that all of the MLPG methods possess excellent rates of convergence, for both the unknown variables and their derivatives. An analysis of computational costs shows that the MLPG5 method is less expensive, both in computational costs as well as definitely in human-labor costs, than the FEM, or BEM. Thus, due to its speed, accuracy and robustness, the MLPG5 method may be expected to replace the FEM, in the near future.

KW - MLPG

KW - MLS

KW - RBF

KW - Test function

KW - Trial function

UR - http://www.scopus.com/inward/record.url?scp=0042964833&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0042964833

VL - 3

SP - 11

EP - 51

JO - CMES - Computer Modeling in Engineering and Sciences

JF - CMES - Computer Modeling in Engineering and Sciences

SN - 1526-1492

IS - 1

ER -