Meshfree methods

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

Space-Time Meshfree Collocation Method for PDEs

Abstract: An innovative Space-Time Meshfree Collocation Method (STMCM) for solving systems of nonlinear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh-based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods which do not have any underlying mesh, but work on a set of nodes only, without an a priori node-to-node connectivity. A regularization technique to overcome the singularity-by-construction and to compute all necessary derivatives of the kernel functions is presented. The method combines the simplicity and straightforwardness of the strong-form computational techniques with the advantages of meshfree methods over the classical ones, especially for coupled engineering problems involving moving interfaces. The key features of the proposed approach are: (i) no need to generate a mesh, (ii) simplified imposition of boundary conditions, (iii) no need to evaluate integral forms of governing equations, (iv) applicability to complex irregularly-shaped domains. The proposed STMCM is applied to linear and nonlinear ordinary and partial differential equations of different types and its accuracy and convergence properties are studied. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Particle integration of element free galerkin method and its application in metal extrusion simulation

Abstract: Element Free Galerkin(EFG) method is very computationally intensive due to the requirement of elegant background cell quadratures.Nodal Integration of Element Free Galerkin(NIEFG) method converts the background cell quadrature into nodal integration,so that it is much more efficient than EFG.However,the existence of zero energy modes in NIEFG results in instability,and some stabilization scheme should be used to stabilize the method which may introduce significant extra errors.Based on the idea of stress points and Newton-Cotes integration,a particle integration of element free Galerkin(PIEFG) method is proposed in this paper.Numerical example of linear elasticity shows that the PIEFG is pretty stable and much more efficient than EFG.Furthermore,PIEFG is extended to the simulation of metal extrusion problems,which shows that PIEFG is very promising in metal extrusion simulation.

Interactive Soft Tissue Deformation Simulation Using Physically Based Modeling

Abstract: This paper presents a study on physically based modeling and simulation of soft tissue deformation, with the goal of producing realistic, real-time effects during the simulation. We consider soft tissue deformation as a solid mechanics problem with a linear elastic constitutive law. A point collocation based meshfree method is employed to solve the governing equations. To achieve real-time performance, an octree data structure is used to organize the support sets and the nodes to expedite the computation in the meshfree method. The developed system brings together the surface representation for visualization and meshfree modeling for physically based animation to set up a virtual reality environment for soft tissue surgery simulation.Copyright © 2006 by ASME

Meshless Method with Radial Basis Functions for Hamilton Canonical Equation

Abstract: Meshless element of Hamilton canonical equation was established in this paper by combining the modified Hellinger-Reissner variational principle for elastic material and radial point interpolation functions. Using Multiquadric (MQ), Gaussian (EXP) and thin plane spine (TPS), the astringency of meshless methods and the effects of the dimensionless shape parameters on the maximum displacement were investigated by the numerical examples of the single or the cross-lay laminated plates. And all of the numerical results of displacement w were compared with that of MSC. Nastran This study introduced the advantages of meshless finite element method into semi-analytic solution of Hamilton canonical equation, and a new semi-analytic method was presented for Hamilton canonical equation.

Solution of Elasto-Statics Problems Using the Element-Free Galerkin Method with Local Maximum Entropy Shape Functions

Abstract: The element free Galerkin method (EFGM) [1] is one of the most robust meshless methods for the solution of elasto-statics problems. In the EFGM, moving least squares (MLS) shape functions are used for the approximation of the field variable. The essential boundary conditions cannot be implemented directly as in the case of Finite Element Method (FEM), because the MLS shape functions do not possess the Kronecker-delta property and use Lagrange multipliers instead. In this paper the recently developed local maximum entropy shape functions are used in the EFGM for the approximation of the field variable instead of MLS. As the local maximum entropy shape functions possess the Kroneckerdelta property at the boundaries so the essential boundary conditions are enforced directly as in the case of FEM. Two benchmark problems, a cantilever beam subjected to parabolic traction at the free end and an infinite plate with circular hole subjected to unidirectional tension are solved to show the implementation and performance of the current approach. The displacement and stresses calculated by the current approach show good agreement with the analytical results.