Meshfree methods

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

Feasibility Of A Meshfree Post-processing For Boundary Element Methods

Abstract: A substantial feature of boundary element methods is that a surface discretization of examined three-dimensional objects suffices to compute the solution of the underlying field problem. However, an auxiliary volume mesh is often created in practical applications to evaluate the solved problem and to apply classical postprocessing methods, for instance to compute and visualize field lines or isosurfaces. In contrast, a meshfree post-processing approach is excellently suitable for boundary element methods but requires adapted and enhanced postprocessing algorithms. Instead of a pre-computation of field values in a previously defined grid or mesh, field values are computed on demand only in points that are really needed to evaluate and to visualize the field. Here, two visualization techniques, the computation of field lines and the computation of isosurfaces, are shown to demonstrate the power of meshfree post-processing approaches using pure boundary element methods. In the case of field line computations, a relatively small number of evaluation points are defined during the determination of a field line based on a Runge-Kutta-Fehlberg method. On the other hand, information about data range of field values must be supplied in the complete space for isovalue search methods. In both cases, a hierarchical octree scheme is applied to group all boundary elements and evaluation points spatially and to provide information about field values in the complete space. The computation of field values is significantly accelerated by a coupling of the post-processing algorithms with the fast multipole method, an established compression technique for boundary element methods. Finally, the numerical examples show that the presented meshfree post-processing approaches enable evaluation and visualization of results of boundary element methods comparable to domain based methods like finite element methods.

Edge meshless method applied to vector electromagnetic problems

Abstract: A challenge in meshless methods dealing with vector electromagnetic problems is to produce numerical solutions that are free of spurious modes given that the generated vector field does not satisfy the condition of zero divergence. The edge meshless method constructs its approximations using specials shape functions based on edges to produce vector fields that are divergence free and to guarantee the continuity of the tangential field component. This work presents the application of the edge meshless method to solve vector electromagnetic problems. An eigenvalue problem is tested to demonstrate that the technique produces correct numerical solution without spurious modes.

Application of Meshless Methods for Solving an Inverse Heat Conduction Problem

Abstract: In this paper, we consider the inverse problem of determining the unknown temperature at x = 0 and section of initial condition at t = 0 in an inverse heat conduction problem (IHCP). Two new numerical methods are developed by using the solution of an auxiliary problem and heat polynomials as basis functions in presence of noisy data. Due to ill-posed IHCP, we use the Tikhonov regularization technique with the GCV scheme to solve the resulting matrix system of the basis function methods (BFM). Some numerical examples are presented to illustrate the strength of the methods.

An Element-Free Galerkin Method Based on Complex Variable Moving Kriging Interpolation for Potential Problems

Abstract: The moving Kriging interpolation (MKI) is an accurate approximation method that has the interpolating property. However, it is rarely used in meshless methods because of its low efficiency. In this paper, we proposed an efficient MKI method, the complex variable moving Kriging interpolation (CVMKI) method, for “domain” type meshless method. Further, we proposed the CVMKI-based element-free Galerkin (CVMKIEFG) method for 2D potential problems. CVMKIEFG is an efficient meshless method and can impose the essential boundary conditions directly and easily. We proposed two formulations for CVMKIEFG: the conventional formulation and the cell-based formulation. The latter formulation is proposed for higher efficiency. Three 2D example problems are presented to demonstrate the efficiency and accuracy of CVMKIEFG.

Impact of different node distributions on the meshless Radial Point Interpolation Method in time-domain electromagnetic simulations

Abstract: This paper investigates the effect of different node distributions on the accuracy of electromagnetic simulations performed with meshless methods. As a test case, a rectangular waveguide truncated with perfectly-matched layers is simulated using the three-dimensional meshless Radial Point Interpolation Method (RPIM) in the time domain. For the discretization of the geometry, different strategies of node distribution are utilized, namely a uniform grid distribution, a cylindrical distribution, and disturbed grid distributions with random displacements amounting to 5% and 10% of the grid average node distance. All distributions are generated with a similar node density, and the results are compared in terms of phase and amplitude of the propagating wave. The application of RPIM in all cases demonstrates similar levels of error, which indicates the robustness of this meshless algorithm with respect to different node distribution strategies.