Meshfree methods

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

Efficient mixed-domain analysis of electrostatic MEMS

Abstract: We present efficient computational methods for scattered point and meshless analysis of electrostatic microelectromechanical systems (MEMS). Electrostatic MEM devices are governed by coupled mechanical and electrostatic energy domains. A self-consistent analysis of electrostatic MEMS is implemented by combining a finite cloud method-based interior mechanical analysis with a boundary cloud method (BCM)-based exterior electrostatic analysis. Lagrangian descriptions are used for both mechanical and electrostatic analyses. Meshless finite cloud and BCMs, combined with fast algorithms and Lagrangian descriptions, are flexible, efficient, and attractive alternatives compared to conventional finite element/boundary element methods for self-consistent electromechanical analysis. Numerical results are presented for MEM switches, a micromirror device, a lateral comb drive microactuator, and an electrostatic comb drive device. Simulation results are compared with experimental and previously reported data for many of the examples discussed in this paper and a good agreement is observed.

The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The Crank–Nicolson scheme and the method of lines (MOL)

Abstract: MHD equations have many applications in physics and engineering. The model is coupled equations in velocity and magnetic field and has a parameter namely Hartmann. The value of Hartmann number plays an important role in the equations. When this parameter increases, using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem. In the present paper a numerical meshless method based on radial basis functions (RBFs) is provided to solve MHD equations. For approximating the spatial variable, a new approach which is introduced by Bozzini et al. (2015) is applied. The method will be used here is based on the interpolation with variably scaled kernels. The methodology of the new technique is defining the scale function c on the domain Ω ⊂ R d . Then the interpolation problem from the data locations x j ∈ R d transforms to the new interpolation problem in the data locations ( x j , c ( x j ) ) ∈ R d + 1 (Bozzini et al., 2015). The radial kernels used in the current work are Multiquadrics (MQ), Inverse Quadric (IQ) and Wendland’s function. Of course the latter one is based on compactly supported functions. To discretize the time variable, two techniques are applied. One of them is the Crank–Nicolson scheme and another one is based on MOL. The numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem. Also in appendix, we provide a computational algorithm for implementing the new technique in MATLAB software.

Dynamic fracture with meshfree enriched XFEM

Abstract: The enrichment of the extended finite element method (XFEM) by meshfree approximations is studied. The XFEM allows for modeling arbitrary discontinuities, but with low order elements the accuracy often needs improvement. Here, the meshfree approximation is used as an enrichment in a cluster of nodes about the crack tip to improve accuracy. Several numerical examples show that this leads to more accuracy for stress intensity factors computations, and to the capability to capture the branching point of a propagating crack from the stresses.

Multi-scale methods

Abstract: In this paper four multiple scale methods are proposed. The meshless hierarchical partition of unity is used as a multiple scale basis. The multiple scale analysis with the introduction of a dilation parameter to perform multiresolution analysis is discussed. The multiple field based on a 1-D gradient plasticity theory with material length scale is also proposed to remove the mesh dependency difficulty in softening/localization problems. A non-local (smoothing) particle integration procedure with its multiple scale analysis are then developed. These techniques are described in the context of the reproducing kernel particle method. Results are presented for elastic-plastic one-dimensional problems and 2-D large deformation strain localization problems to illustrate the effectiveness of these methods. Copyright © 2000 John Wiley & Sons, Ltd.

A meshfree weak-strong (MWS) form method for time dependent problems

Abstract: A meshfree weak-strong (MWS) form method, which is based on a combination of both the strong form and the local weak form, is formulated for time dependent problems. In the MWS method, the problem domain and its boundary are represented by a set of distributed field nodes. The strong form or the collocation method is used to discretize the time-dependent governing equations for all nodes whose local quadrature domains do not intersect with natural (derivative or Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The natural boundary conditions can then be easily imposed to produce stable and accurate solutions. The moving least squares (MLS) approximation is used to construct the meshfree shape functions in this study. Numerical examples of the free vibration and dynamic analyses of two-dimensional structures as well as a typical microelectromechanical system (MEMS) device are presented to demonstrate the effectivity, stability and accuracy of the present MWS formulation.