Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Towards the development of unconditionally stable time-domain meshless numerical methods

Abstract: Meshless methods have recently emerged as robust numerical techniques for electromagnetic modeling in time domain. In those methods, a problem domain is represented by scattered spatial nodes instead of numerical meshes, thus the conformal modeling of boundaries and solution refinements can be conveniently achieved. However, the CFL-like numerical stability condition still exists with these meshless methods, which prevents the methods being efficiently applied for general electromagnetic simulations. To overcome the problem, in this paper, we propose the unconditionally stable mesheless methods by incorporating two efficiency-improved implicit schemes, namely the leapfrog alternating-direction-implicit (ADI) and the locally one-dimensional scheme (LOD) schemes, into the radial point interpolation mesheless method (RPIM). The proposed methods are numerically verified for their unconditional stability, and are assessed for their numerical accuracy and efficiency. In comparisons with the conventional RPIM, computational cost can be saved by up to 80% with little sacrifice of accuracy.

Groundwater management using a new coupled model of meshless local Petrov-Galerkin method and modified artificial bee colony algorithm

Abstract: To develop sustainable groundwater management strategies, generally coupled simulation-optimization (SO) models are used. In this study, a new SO model is developed by coupling moving least squares (MLS)-based meshless local Petrov-Galerkin (MLPG) method and modified artificial bee colony (MABC) algorithm. The MLPG simulation model utilizes the advantages of meshless methods over the grid-based techniques such as finite difference (FDM) and finite element method (FEM). For optimization, the basic artificial bee colony algorithm is modified to balance the exploration and exploitation capacity of the model more effectively. The performance of the developed MLPG-MABC model is investigated by applying it to hypothetical and field problems with three different management scenarios. The model results are compared with other available SO model solutions for its accuracy. Further, sensitivity analyses of various model parameters are carried out to check the robustness of the SO model. The proposed model gave quite promising results, showing the applicability of the present approach.

Development of two dimensional groundwater flow simulation model using meshless method based on MLS approximation function in unconfined aquifer in transient state

Abstract: In recent decades, due to reduction in precipitation, groundwater resource management has become one of the most important issues considered to prevent loss of water. Many solutions are concerned with the investigation of groundwater flow behavior. In this regard, development of meshless methods for solving the groundwater flow system equations in both complex and simple aquifers9 geometry make them useful tools for such investigations. The independency of these methods to meshing and remeshing, as well as its capability in both reducing the computation requirement and presenting accurate results, make them receive more attention than other numerical methods. In this study, meshless local Petrov–Galerkin (MLPG) is used to simulate groundwater flow in Birjand unconfined aquifer located in Iran in a transient state for 1 year with a monthly time step. Moving least squares and cubic spline are employed as approximation and weight functions respectively and the simulated head from MLPG is compared to the observation results and finite difference solutions. The results clearly reveal the capability and accuracy of MLPG in groundwater simulation as the acquired root mean square error is 0.757. Also, with using this method there is no need to change the geometry of aquifer in order to construct shape function.

The numerical simulation of crack propagation using radial point interpolation meshless methods

Abstract: In this work, a new crack propagation prediction algorithm is combined with two distinct meshless methods – the Radial Point Interpolation Method (RPIM) and the Natural Neighbor Radial Point Interpolation Method (NNRPIM). Both these advanced discretization numerical techniques construct the shape functions using the radial point interpolating technique, which allows to build shape functions possessing the delta Kronecker property. The RPIM requires a background integration grid to numerically integrate the partial differential equations ruling the physical phenomenon. Alternatively, the NNRPIM constructs the background integration points using only the information of the spatial position of the nodes. This work aims to present a new methodology to predict the crack propagation and, simultaneously, to compare the performance of both meshless techniques (for similar conditions). The developed algorithm advances the crack tip iteratively and uses the maximum tangential stress criterion to calculate the propagation direction. In the end, three benchmark tests were analyzed using the proposed algorithm. Accurate crack paths were obtained when compared to the solutions described in the literature.

Direct meshless local Petrov–Galerkin method for the two-dimensional Klein–Gordon equation

Abstract: In this paper we apply the direct meshless local Petrov–Galerkin (DMLPG) method to solve the two dimensional Klein–Gordon equations in both strong and weak forms. Low computational cost is the main property of this method compared with the original MLPG technique. The reason lies behind the approach of generalized moving least squares approximation where the discretized functionals, obtained from the PDE problem, are directly approximated from nodal values. This shifts the integration over polynomials rather than the MLS shape functions, leading to an extremely faster scheme. We will see that this method can successfully solve the problem with a reasonable accuracy.