Meshfree methods

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

Meshless Interpolating Moving Least Square Mixed Collocation Method for Elasticity Problem

Abstract: In this paper, a new interpolating moving least square (IMLS) method is proposed which has the properties of moving least square (MLS) and the radial basis function (RBF). In the approach, the polynomial and the RBF are combined. The coefficients of the polynomial are obtained by MLS approximation and the coefficients of RBF are obtained through the condition that the interpolation function obtained passes through all scattered nodes in an influence domain. Thus shape functions preserve the Kronecker delta function property which makes the implementation of essential boundary conditions much easier than the meshless methods based on the MLS approximation. In addition, the diffuse derivatives instead of full derivative are applied for the derivatives of shape functions. Thus, the computational efficiency is improved. Examples on elasticity problems show that the present method has the same accuracy and convergence rate as MLS.

SPH modeling for soil mechanics with application to landslides

Abstract: Thanks to its meshfree feature, the smoothed particle hydrodynamics (SPH) has apparent advantages over the traditional finite element method for addressing large deformation problems in geotechnical engineering, eg, landslides This chapter provides the essentials of SPH theory and the key ingredients of SPH modeling Especially, the stability-related issues are investigated and discussed by carrying out some benchmarking examples Then, the effectiveness of SPH modeling is validated by applying it to a series of granular column collapse and earthquake-induced landslides Finally, the most recent development of SPH is reviewed, and the Pros/Cons of SPH modeling relative to the conventional grid-based methods and other meshfree methods are explored in detail Some latest advice on the use of SPH in geotechnical engineering is also involved in this chapter

A meshless multiscale method for simulating hemodynamics

Abstract: Simulating hemodynamic quantities such as pressure and velocity are of great interest to clinicians to aid in surgical planning. To accurately simulate modifications to a region of vasculature, the entire system must be modeled. To facilitate this, a localized Radial-Basis Function (RBF) collocation Meshless flow solver is developed and tightly coupled to a 0D Lumped-Parameter Model (LPM) for solution of the peripheral circulation. The Meshless solver uses localized RBF collocations at data points that are automatically generated according to the geometry. The incompressible flow equations are updated by an explicit time-marching scheme based on a pressure-velocity correction algorithm. The inlets and outlets of the domain are tightly coupled with the LPM which contains elements that draw from a fluid-electrical analogy such as resistors, capacitors, and inductors that represent the viscous resistance, vessel compliance, and flow inertia, respectively. The localized RBF Meshless approach is well-suited for modeling complicated non-Newtonian hemodynamics due to ease of spatial discretization, ease of addition of multi-physics interactions such as fluid-structure interaction of the vessel wall, and ease of parallelization for fast computations. This work introduces the tight coupling of Meshless methods and LPMs for fast and accurate hemodynamic simulations.

A linear smoothed meshfree method with intrinsic enrichment functions for 2D crack analysis

Abstract: PurposeThis article aims to develop an accurate and efficient meshfree Galerkin method based on the strain smoothing technique for linear elastic continuous and fracture problems.Design/methodology/approachThis paper proposed a generalized linear smoothed meshfree method (LSMM), in which the compatible strain is reconstructed by the linear smoothed strains. Based on the idea of the weighted residual method and employing three linearly independent weight functions, the linear smoothed strains can be created easily in a smoothing domain. Using various types of basic functions, LSMM can solve the linear elastic continuous and fracture problems in a unified way.FindingsOn the one hand, the LSMM inherits the properties of high efficiency and stability from the stabilized conforming nodal integration (SCNI). On the other hand, the LSMM is more accurate than the SCNI, because it can produce continuous strains instead of the piece-wise strains obtained by SCNI. Those excellent performances ensure that the LSMM has the capability to precisely track the crack propagation problems. Several numerical examples are investigated to verify the accurate, convergence rate and robustness of the present LSMM.Originality/valueThis study provides an accurate and efficient meshfree method for simulating crack growth.

Design of boundary interpolatable moving least squares approximation

Abstract: Most of the meshless or mesh-free methods rely on the meshless approximation methods. Through the meshless approximations, the shape functions for unknown variables, which are utilized in variational weak forms of ordinary or partial differential equations, can be constructed only with the nodal points with no aid of well-defined mesh. The feature of independency of meshes in constructing the shape functions gives various potentials in dealing with engineering problems. And the drawbacks encountered in applying the finite element methods, such as human labor-intensive meshing, degradation of solution accuracy according to the element distortions, burden of re-meshing during large deformations, locking, and others, have been expected to be eliminated or alleviated by adopting the meshless or mesh-free approach. With the anticipation, considerable research efforts have been given to the field of meshless approximation and its application to the analysis of engineering problems. As a result, various meshless analysis methods have been proposed to alleviate the drawbacks of mesh (or grid) dependent conventional analysis methods such as finite element or finite difference methods. They are SPH(smoothed particle hydrodynamics), EFG(element free Galerkin method), Generalized Finite Difference, Generalized Finite Element, MLPG(meshless local Petrov-Galerkin method), and others[1]. Even though those have their own salient features, all of them utilize the nodal shape functions obtained from the meshless(or mesh-free) approximation schemes for the weak forms or differential equations. Consequently all of the meshless analysis methods inherit the common character from the meshless approximation, and the character makes it possible to eliminate or alleviate the mesh-related drawbacks in the conventional numerical analysis methods such as finite element or finite difference methods. The most representative meshless approximation scheme is the moving least squares method[2]. Its lowest version is the same as Shepard interpolation, and it is essentially the same as the RKPM(reproducing kernel particle method)[1]. For its generalized version, one can see reference[3]. However, due to the diffuse character of meshless approximations including the moving least squares scheme, the meshless approximations lose the exact interpolation property, differing from the Lagrange interpolation functions. And the lack of exact interpolation property leads to the difficulty in enforcement of essential boundary conditions when we apply the mesh less methods, such as SPH, EFG, GFDM, GFEM, MLPG, and others, in solving the partial differential equations, although the treatment of essential boundary conditions is very easy in finite element method. Although several approaches have been proposed to resolve this unexpected trouble related to treatment of essential boundary conditions, the trouble is not eliminated completely. Some of these are the Lagrange multiplier method, the penalty method, the mixed (partial) transformation method, and so on[1]. Therefore, most of the meshless methods based on mesh less approximation scheme still have troubles and inconveniences in enforcing the essential boundary conditions. Under this background, this work aims to completely eliminate the trouble in enforcement of essential boundary conditions in mesh less methods, focusing on the moving least squares method. For the purpose, by using the newly devised metric function and weight functions, a novel BI(boundary interpolatable) moving least squares scheme is proposed, where the constructed mesh less approximated function has the exact interpolation property along the boundary. And consequently, the trouble concerning the essential boundary condition has naturally disappeared due to the exact interpolation property along the boundary.