Meshfree methods

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

Iterative meshfree methods for the Helmholtz equation

Abstract: For acoustic computations in the mid-frequency range the finite element method (FEM) is a well-known standard tool. Unfortunately, for increasing frequencies, i.e. higher wavenumbers, the FEM suffers from the so-called pollution effect which is mainly a consequence of the dispersion, meaning that the numerical wavenumber and the exact wavenumber disagree. Using meshfree methods as, e.g., the radial point interpolation method (RPIM) or the element-free Galerkin method (EFGM) can reduce this effect significantly. Moreover, meshfree methods allow the usage of shape functions that can be adapted to the differential equation to be solved. Consequently, an iterative method can be derived, which uses a standard meshfree method to compute a first approximation for the given problem. In a second step this approximation is taken to construct new shape functions that are able to better reproduce the wave-like character of the solution. If a few requirements are considered, this method leads to better results in a more efficient way. In this paper two meshfree methods, namely an iterative RPIM and an iterative EFGM, are examined. The methods are compared to the FEM and restrictions for an efficient applicability are shown.

Evaluation of a radial basis function node refinement algorithm applied to bioheat transfer modeling

Abstract: Many bioheat transfer problems involve linear/non-linear equations with non-linear or time-dependent boundary conditions. For heat transfer problems, the presence of time and space-dependent functions under Neumann and Mixed type boundary conditions characterize trivial applications in bioengineering, such as thermotherapies, laser surgeries, and burn studies. This greatly increases the complexity of the numerical solution in several problems, requiring fast and accurate numerical solutions. This paper has a main objective evaluate an adaptive mesh refinement radial basis function method strategy for the classical Penne's bioheat transfer modeling. Our numerical results had errors of ~0.1% compared to analytical solutions. Thus, the proposed methodology is accurate and has a low computational cost. For step function heating, two RBF shape parameters were applied, again achieving excellent results. The distributions of the nodes in the solution domain show that the primary source of error in the numerical solutions came from the boundary conditions. This finding should arouse the interest of engineers and scientists in the development of new strategies for problems involving boundary conditions with periodic functions.

A meshfree method with domain decomposition for Helmholtz boundary value problems

Abstract: In the framework of meshfree methods, we address the numerical solution of boundary value problems (BVP) for the non-homogeneous modified Helmholtz partial differential equation (PDE). In particular, the unknown solution of the BVP is calculated in two steps. First, a particular solution of the PDE is approximated by superposition of plane wave functions with different wavenumbers and directions of propagation. Then, the corresponding homogeneous BVP is solved, for the homogeneous part of the solution, using the classical method of fundamental solutions (MFS). The combination of these two meshfree techniques shows excellent numerical results for non-homogeneous BVPs posed in simple geometries and when the source term of the PDE is sufficiently regular. However, for more complex domains or when the source term is piecewise defined, the MFS fails to converge. We overcome this problem by coupling the MFS with Lions non-overlapping domain decomposition method. The proposed technique is tested for the modified Helmholtz PDE with a discontinuous source term, posed in an L-shaped domain.

Nonlocal Damage Modelling Using the Element-free Galerkin Method in the Frame of Finite Strains

Abstract: Computational analysis of damage failure is of great importance in predicting assessment of structure integrity. Numerical modeling of ductile material damage using finite element methods often suffers from convergence problems of numerical iteration, especially, when working with a complex constitutive model as gradient plasticity and nonlocal damage models. Due to large strains in damaging elements the computation may result in non-convergence. For the higher order gradient plasticity the element formulation is often necessary, which causes additional difficulties in implementation and computations. In recent years meshless methods have been developed as an alternative for the finite element method (FEM) and can overcome some known shortcomings of the latter. One major advantage of the meshless methods is in continuous differentiation of the strain tensor for cases with finite strains. Complex constitutive models, such as gradient plasticity nonlocal damage models, are easy to be applied in meshless methods. In the present paper we have developed and implemented an algorithm of element-free Galerkin (EFG) methods for straingradient based nonlocal damage models and used it to simulate ductile material damage. The method provides a reliable and robust method for material failure with large damage zones. With the help of the meshless method material failure of specimens as well as the size effect are predicted accurately.

A Meshless Approach to Radionuclide Transport Calculations

Abstract: Over the past thirty years numerical modelling has emerged as an interdisciplinary scientific discipline which has a significant impact in engineering and design. In the field of numerical modelling of transport phenomena in porous media, many commercial codes exist, based on different numerical methods. Some of them are widely used for performance assessment and safety analysis of radioactive waste repositories and groundwater modelling. Although they proved to be an accurate and reliable tool, they have certain limitations and drawbacks. Realistic problems often involve complex geometry which is difficult and time consuming to discretize. In recent years, meshless methods have attracted much attention due to their flexibility in solving engineering and scientific problems. In meshless methods the cumbersome polygonization of calculation domain is not necessary. By this the discretization time is reduced. In addition, the simulation is not as discretization density dependent as in traditional methods because of the lack of polygon interfaces. In this work fully meshless Diffuse Approximate Method (DAM) is used for calculation of radionuclide transport. Two cases are considered; First 1D comparison of 226 Ra transport and decay solved by the commercial Finite Volume Method (FVM) and Finite Element Method (FEM) based packages and DAM. This case shows the level of discretization density dependence. And second realistic 2D case of near-field modelling of radionuclide transport from the radioactive waste repository. Comparison is made again between FVM based code and DAM simulation for two radionuclides: Long-lived 14 C and short-lived 3 H. Comparisons indicate great capability of meshless methods to simulate complex transport problems and show that they should be seriously considered in future commercial simulation tools. Numerical simulations of convective diffusive transport partial differential equations are widely used for the solution of many engineering and scientific problems. Traditional numerical methods i.e. Finite Difference Method (FDM), FVM, FEM are well accepted and tested on numerous problems. Nowadays a new branch of Meshless Numerical Methods (MSM) is in fast theoretical development and is also progressively used for the solution of practical engineering problems. Numerical modelling is essential in particular for the prediction of long-term processes such as for example toxic releases in the environment. Long time-frames involved in the movement of substances especially in the ground are caused by the slow movement of the groundwater. Slow and long-term processes are not an easy task to solve numerically. The