Meshfree methods

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

On meshless methods : a novel interpolatory method and a GPU-accelerated implementation

Abstract: Meshless methods have been developed to avoid the numerical burden imposed by meshing in the Finite Element Method. Such methods are especially attrac- tive in problems that require repeated updates to the mesh, such as problems with discontinuities or large geometrical deformations. Although meshing is not required for solving problems with meshless methods, the use of meshless methods gives rise to different challenges. One of the main challenges associated with meshless methods is imposition of essential boundary conditions. If exact interpolants are used as shape functions in a meshless method, imposing essen- tial boundary conditions can be done in the same way as the Finite Element Method. Another attractive feature of meshless methods is that their use involves compu- tations that are largely independent from one another. This makes them suitable for implementation to run on highly parallel computing systems. Highly par- allel computing has become widely available with the introduction of software development tools that enable developing general-purpose programs that run on Graphics Processing Units. In the current work, the Moving Regularized Interpolation method has been de- veloped, which is a novel method of constructing meshless shape functions that achieve exact interpolation. The method is demonstrated in data interpolation and in partial differential equations. In addition, an implementation of the Element-Free Galerkin method has been written to run on a Graphics Processing Unit. The implementation is described and its performance is compared to that of a similar implementation that does not make use of the Graphics Processing Unit.

Study on meshless manifold method in elastodynamics

Abstract: The meshless manifold method is utilized to analyze transient deformations of the elastodynamics,especially,with discontinuity in the solving domain.The shape function is built with the method by the partition of unity and the finite cover technology,so the shape function cannot be effected by discontinuity in the domain to treat continuous and discontinuous dynamics problem easily.To local problemst,he shape functions are built more effectively than other method.So the method can avoid the disadvantages in other meshless methods in which the tip of the discontinuous crack isn′t considered.The approximation functions will not be influenced by the discontinuity in the solving domain if finite cover technology is employed in the method,which can overcome some difficulties when the problems are solved with the meshless methods.When the meshless manifold method is utilized to analyze the elastodynamics,the method is divided into the major two parts:the discrete space of the domain is used to the partition of unity,which the present method void to mesh element and refine element.So the method has good continuity in the area of the computing stress for element-free of the analysis problem.The Newmark methods is used for the time integration scheme.The scheme is a widely direct integration method.The local weak formulation of the dynamic partial differential equation for elastic is derived from the method of weighted residuals(MWR).At last,the validity and accuracy of the presented meshless manifold method solution are illustrated by the 2D plate of the elastodynamics.The meshless manifold method results show that the stresses and the displacements at the critical point agree well with those obtained from the analytical solution.

Numerical analysis of 2-D linear elastic problems by MLPG method

Abstract: Meshless methods have attracted considerable attention of the scientific community over the last two decades due to their flexibility and high continuity of meshless approximation functions. However, high numerical costs in comparison to the Finite Element (FE) Method still impede their wide commercial use. In addition, primal meshless methods are in general plagued by various locking phenomena, similarly to the comparable FE formulations. Collocation meshless methods are computationally more efficient than those based on the integration of various weak forms, but they suffer from instability and lack of accuracy, caused primarily by the problematic imposition of Neumann boundary conditions (BCs). The mixed Meshless Local Petrov-Galerkin (MLPG) Method paradigm represents an efficient remedy for these deficiencies. So far it has been successfully applied for solving various engineering problems, such as the bending of thin beams, plates and shells, the topology-optimization, or electrodynamics. It is computationally superior to the primal meshless approaches because the differentiation of meshless functions in the entire domain is avoided, which increases numerical efficiency and stability. Furthermore, the mixed approach decreases the continuity requirements of the trail functions and allows the use of the shape functions of a lower order. This enables the use of smaller support domains of shape functions and further decreases computational costs. This contribution deals with the application of the MLPG approach in two-dimensional (2-D) linear elasticity, whereby special attention is dedicated to the mixed strategy. Various formulations based on either a weak or strong form of governing equations are presented. Discretization is performed by a set of nodes that are not connected into the mesh of elements. The local weak form (LWF) of the equilibrium equation is derived over the prismatic local sub-domains that surround the nodes. In the case of the collocation formulations, the Dirac Delta functions are applied at each nodal point as the test functions in LWF. The stress and displacements components are approximated independently by using the same trail shape function. Various approximation techniques are used, including the Moving Least Squares (MLS) functions, the Polynomial Point Interpolation Method (PPIM), the multi-quadrics Radial Basis Functions (MQ-RBF) and the B-splines functions. The approximated stresses are used to discretize the strong or weak forms of the equilibrium equations. The stress nodal values are then eliminated from the equations by enforcing the compatibility between the interpolated stresses and displacements at the nodes via the collocation, which yields a global system of equations with only nodal displacements as unknowns. In the collocation, special attention is dedicated to the imposition of the Force BCs, which is performed by the direct collocation and the penalty method. The performance of the presented algorithms is compared by few numerical examples.

Dual Reciprocity Hybrid Boundary Node Method for Solving Poisson Equations

Abstract: It has long been claimed that the boundary element method (BEM) is a viable alternative to the domain-type finite element method (FEM) and finite difference method (FDM) due to its advantages in dimensional reducibility and suitability to infinite domain problems. However, it has a major difficulty in handling inhomogeneous terms such as time-dependent and nonlinear problems. It is over 18 years since the dual reciprocity method (DRM) was first proposed by Nardini and Brebbia which provides a very general methodology for obtaining a boundary element solution to wide range of problems. Another obstacle in BEM, just like the FEM, surface mesh or remesh requires costly computation, especially for moving boundary and nonlinear problems. The boundary-type meshless methods such as the hybrid boundary node method (Hybrid-BNM) , the boundary node method (BNM) shown an emerging technique to alleviate these drawbacks.