A simple and less-costly integration of meshless Galerkin weak form

Computational efficiency and reliability are the most important requirements for the success of a meshless numerical technique. The efficiency of these methods depends on the proper choice of the interpolation scheme, numerical integration procedures and techniques of imposing the boundary conditions. From all these difficulties, the integration is the most time-consuming part in meshless calculation due to the large number of integration points needed for a sufficiently accuracy of the integration of the weak form. Also, insufficiently accurate numerical integration may lead a deterioration and rank-defieciency in the numerical solution. The difficulty is due to the complexity of the MLS shape functions in an integration domain.

The purpose of the present paper is to alleviate the difficulty in the numerical integration of the weak form in the MLPG [

1

] and EFG [

2

] methods.

For this aim we implement a 3D integration technique for the evaluation of the stiffness matrix that does not rely on a partition of the domain into cells. This is made by using Quasi-Monte Carlo techniques [

3

]. The integration domain can be the entire computational domain, without the need to use background meshes.

For comparison of the efficiency and accuracy for these two meshless formulations based on Galerkin weak form the study is based on various Quasi-Monte Carlo sequences.

The method is applicable to any type of problem with any number of dimensions. Here, we have presented a numerical test for a 3D elasticity problem. The test demonstrates the efficiency of the integration techniques for both meshless formulations.

We find this new method simultaneously simple and efficient because the integration scheme is related to the set of quasi random points used for the approximation. We also expect that our method will be found especially simple for 3D problems with complicated shapes.