Algoritmos eficientes em métodos sem malha - Alexandre Ramos Fonseca (2011)
Unlike mesh based methods, meshfree methods are distinguished by the use of a set of scattered nodes over the domain instead of a mesh or grid. In this work the use of MLPG (Meshless Local Petrov Galerkin method) to solve electromagnetic problems is discussed. The main objective is to improve the methods computational efficiency in order to make it competitive with traditional methods like the finite element method. Due to the absence of a mesh there is no connectivity among nodes and it is a challenge for the method to determine efficiently which nodes belong to the neighborhood of a particular node. To solve this problem a kd-tree is used. Another issue of meshfree methods is how to impose Dirichlet boundary conditions when the shape functions do not possess the Kronecker delta property, which is the case for shapefunctions constructed from the moving least-squares method (MLS). The MLS is an efficient method, well known in the literature and one of the most used in meshless methods but requires special techniques to impose the Dirichlet boundary conditions, such as the method of penalties or Lagrange multipliers. In addition to the MLS, we investigate shape functions based on the point interpolation method (PIM) using radial basis functions (RPIM) with polynomial terms (RPIMp). The RPIMp has the property of Kronecker delta, which makes it an alternative to the MLS dispensing special techniques for imposing boundary conditions. However, RPIMp presents a higher computational cost than the MLS when a large number of nodes is used. In this paper we propose a mixed method that combines RPIMp to border nodes and the MLS into the domain. Thus, we generate a method that combines the best attributes of two functions: a direct imposition of boundary conditions and fast processing. Results show that the mixed method has good accuracy and intermediate computational cost between the shape functions used.To make the mixed method even more attractive, we investigated a simpler shape function to use inside the domain. Shepard functions (particular case of MLS with C0 consistency) are extremely simple functions, easy to implement and have low computational cost. In this case, even constructing the shape functions with few neighboring nodes, the mixed method always present lower computational cost when compared to using only RPIMp functions. During MLPG implementations we realize that the nodes contributions for the global matrix system are independent. Each node contributes with a line on the matrix system, not interfering with the contribution of the other nodes. Taking advantage of multi-core processors, now easy to find in the market, we propose a way to parallelize the linear system assembling process which is the stage of the method with higher computational cost. Results indicate a performance gain up to 3.78 times in this part of the processing for 4-cores processors