Meshfree methods

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

Linear Solvers for the Finite Pointset Method

Abstract: Many simulations in Computational Engineering suffer from slow convergence rates of their linear solvers. This is also true for the Finite Pointset Method (FPM), which is a Meshfree Method used in Computational Fluid Dynamics. FPM uses Generalized Finite Difference Methods (GFDM) in order to discretize the arising differential operators. Like other Meshfree Methods, it does not involve a fixed mesh; FPM uses a point cloud instead. We look at the properties of linear systems arising from GFDM on point clouds and their implications on different types of linear solvers, specifically focusing on the differences between one-level solvers and Multigrid Methods, including Algebraic Multigrid (AMG). With the knowledge about the properties of the systems, we develop a new Multigrid Method based on point cloud coarsening. Numerical experiments show that our Multicloud method has the same advantages as other Multigrid Methods; in particular its convergence rate does not deteriorate when refining the point cloud. In future research, we will examine its applicability to a broader range of problems and investigate its advantages in terms of computational performance.

Meshfree Analysis of Elastic Bar

Abstract: This thesis presents the application of Element Free Galerkin method for analysis of an Elastic Bar The Element Free Galerkin method is one of the efficient mesh free methods which needs only nodal data and does not need any nodal connectivity, which provides an advantage over the FEM's The need for the meshfree methods arose from the fact that the mesh based methods employ the elements for the imposition of the field values These elements when distorted or misaligned with the back ground mesh give faulty and inaccurate results, further the simulation of explosion or the crack growth is very difficult in FEM's The results presented in this thesis are checked for validity and accuracy of numerical solutions and compared with the exact solution Convergence study has been performed Results show that the proposed formulations are able to reproduce the exact solution with high accuracy which indicates that these methods would be suitable in more complex environments and domains-settings

A meshless manifold method for crack propagation analysis

Abstract: A meshless manifold method (MMM) is presented to analyze the problems of crack propagation, especially the advantages of MMM for irregular cracks. The shape functions in this method are formed by the partition of unity and the finite cover technology,so the shape functions are not affected by discontinuous domains and crack problems can be more properly treated. For strain localization problems,the shape functions can be more effectively established,compared with other methods in which the tips of the discontinuous cracks are not considered. Compared with the conventional numerical manifold method,the shapes of the finite covers can be selected more easily. The finite covers and the partition of unity functions are formed by using the influence domains of a series of nodes with an advantages over the mesh-based numerical manifold method. Compared with the conventional meshless methods,the test functions are not influenced by the discontinuities in the solution domain since finite cover technology is used to overcome some difficulties inherented in the conventional meshless methods. In this paper,the meshless manifold method (MMM) is applied to analyze crack growth in rock samples. The weak solution of the partial differential equation for elasticity are derived using the method of weighted residuals (MWR). Finally,a problem with crack growth under complex stress state is solved with the MMM,the numerical results agree well with the test data,and the validity and accuracy of the MMM are demonstrated.

Three-dimensional FE-EFGM adaptive coupling with application to nonlinear adaptive analysis

Abstract: Three-dimensional problems with both material and geometrical nonlinearities are of practical importance in many engi- neering applications, e.g. geomechanics, metal forming and biomechanics. Traditionally, these problems are simulated using an adaptive finite element method (FEM). However, the FEM faces many challenges in modeling these problems, such as mesh distortion and selection of a robust refinement algorithm. Adaptive meshless methods are a more recent technique for modeling these problems and can overcome the inherent mesh based drawbacks of the FEM but are com- putationally expensive. To take advantage of the good features of both methods, in the method proposed in this paper, initially the whole of the problem domain is modeled using the FEM. During an analysis those elements which violate a predefined error measure are automatically converted to a meshless zone. This zone can be further refined by adding nodes, overcoming computationally expensive FE remeshing. Therefore an appropriate coupling between the FE and the meshless zone is vital for the proposed formulation.One of the most widely used meshless methods, the element-free Galerkin method (EFGM), is used in this research. Maximum entropy shape functions are used instead of the conventional moving least squares based formulations‘. These shape functions posses a weak Kronecker delta property at the boundaries of the problem domain, which allows the essential boundary conditions to be imposed directly and also helps to avoid the use of a transition region in the coupling between the FE and the EFG regions. Total Lagrangian formulation is preferred over the update Lagrangian formulation for modeling finite deformation due to its computational efficiency. The well-established error estimation procedure of Zienkiewicz-Zhu is used in the FE region to determine the elements requiring conversion to the EFGM. The Chung and Belytschko error estimator is used in the EFG region for further adaptive refinement. Numerical examples are presented to demonstrate the performance of the current approach in three dimensional nonlinear problems.

SUPG stabilization of meshfree methods for convection‐dominated problems

Abstract: Streamline-Upwind/Petrov-Galerkin (SUPG) stabilization of meshfree methods with stabilization parameters calculated from standard formulas, which have their roots in mesh-based methods, often leads to unsatisfactory results. A new procedure for obtaining adequate SUPG stabilization parameters for meshfree methods is presented. It is shown that the Kronecker-δ property is very advantageous for a successful stabilization because this leads to local stabilization criteria. If the meshfree shape functions lack this property, a stabilization based on local properties – i.e. on nodes in the neighbourhood – cannot be sufficient for highly convection-dominated problems.