Meshfree methods

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

Meshfree Method Analysis of Biot's Consolidation Using Cell-Based Smoothed Point Interpolation Method

Abstract: A set of cell-based smoothed point interpolation methods are proposed for the numerical analysis of Biot’s formulation. In the proposed methods, the problem domain is discretized using a triangular background mesh. Shape functions are constructed using either polynomial or radial point interpolation method (PIM), leading to the delta function property of shape functions and consequently, easy implementation of essential boundary conditions. The Biot’s equations are discretised in space and time. A variety of support domain selection schemes (T-schemes) are investigated. The accuracy and convergence rate of the proposed methods are examined by comparing the numerical results with the analytical solution for the benchmark problem of one dimensional consolidation.

On the infinite particle limit in Lagrangian dynamics and convergence of optimal transportation meshfree methods

Abstract: We consider Lagrangian systems in the limit of infinitely many particles. It is shown that the corresponding discrete action functionals Gamma-converge to a continuum action functional acting on probability measures of particle trajectories. Also the convergence of stationary points of the action is established. Minimizers of the limiting functional and, more generally, limiting distributions of stationary points are investigated and shown to be concentrated on orbits of the Euler-Lagrange flow. We also consider time discretized systems. These results in particular provide a convergence analysis for optimal transportation meshfree methods for the approximation of particle flows by finite discrete Lagrangian dynamics.

Kernel-based Approximation Methods for Generalized Interpolations: A Deterministic or Stochastic Problem?

Abstract: In this article, we solve a deterministically generalized interpolation problem by a stochastic approach. We introduce a kernel-based probability measure on a Banach space by a covariance kernel which is defined on the dual space of the Banach space. The kernel-based probability measure provides a numerical tool to construct and analyze the kernel-based estimators conditioned on non-noise data or noisy data including algorithms and error analysis. Same as meshfree methods, we can also obtain the kernel-based approximate solutions of elliptic partial differential equations by the kernel-based probability measure.

A Meshfree Method with Fundamental Solutions for Inhomogeneous Elastic Wave Problems

Abstract: We consider the numerical solution of the inhomogeneous Cauchy-Navier equations of elastodynamics, assuming time-harmonic variation for the displacement field U(x;t) = u(x)e i!t of an isotropic material with Lam´ e constants and and density: The resulting elliptic PDE, posed in a bounded simply connected domain is coupled with Dirichlet boundary conditions and solved trough a meshfree method, based on the Method of Fundamental Solutions (MFS). ( u + ( + )r(r u) +! 2 u = f in u = g on In particular, an extension, from the scalar [1, 2] to the vector case, of the MFS is applied and the displacement field u is approximated in terms of a linear combination of fundamental solutions (Kupradze tensors) of the corresponding homogeneous PDE with different source points and test frequencies. The applicability of the numerical method is justified in terms of density results [3]. The high accuracy and the convergence of the proposed method will be illustrated through 2D numerical simulations. Convex and non-convex domains and different sets of boundary data and body forces will be considered. Interior elastic wave scattering problems will also be addressed.

Parallel Computation of Meshfree Methods for Extremely Large Deformation Analysis

Abstract: Due to the heavier computation requirement than other competitive techniques and the essence of applications that are usually highly complex and computationally intensive, parallel computing is especially attractive for these meshfree methods.We only focus on the Reproducing Kernel Particle Method (RKPM), one of the meshfree methods for large strain elasto-plastic analysis of solid and structures, in considering with its ability to accurately model extremely large deformations without mesh distortion problems, and its ease of adaptive modeling by simply changing particle definitions for desired refinement regions. The parallel procedure primarily consists of a mesh partitioning pre-analysis phase, and a parallel analysis phase that includes explicit message passing among partitions on individual processors. With redefinition techniques applied to the shared zones of different geometrical parts, the graph-based procedure Metis, which is quite popular for mesh-based analysis, is used for partitioning in this meshfree analysis. Parallel simulations have been conducted on an SGI Onyx3900 supercomputer with MPI message passing statements. The effectiveness and performance with different partitions has then been compared, and a comparison of the meshfree method with finite element methods is also presented.