Meshfree methods

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

Influence of size effect on flapwise vibration behavior of rotary microbeam and its analysis through spectral meshless radial point interpolation

Abstract: In this paper, by utilizing the modified couple stress theory, influence of size effect on flapwise vibration behavior of rotary microbeam is analyzed based on Bernoulli–Euler beam model. It should be mentioned that this theory contains the size effect by considering the material length scale parameter unlike the classical continuum theories. Governing equation and boundary conditions are derived by using Hamilton’s principle. Governing equation is solved for cantilever and propped cantilever boundary conditions by applying spectral meshless radial point interpolation (SMRPI) method. SMRPI is based on meshless methods and benefits from spectral collocation ideas. The point interpolation method with the help of radial basis functions is proposed to construct shape functions which have Kronecker delta function property. Evaluation of high order derivatives is possible by constructing and using operational matrices. The computational cost of the method is modest due to using strong form equation and collocation approach.

Edge Meshless Method Applied to Vector Electromagnetic Problems

Abstract: A challenge in meshless methods dealing with vector electromagnetic problems is to produce numerical solutions that are free of spurious modes given that the generated vector field does not satisfy the condition of zero divergence. The edge meshless method (EMM) constructs its approximations using special shape functions based on edges to produce vector fields that are divergence free and to guarantee the continuity of the tangential field components. This paper presents the application of the EMM to solve vector electromagnetic problems. The 2-D Maxwell eigenvalue problem with anisotropic medium is tested to demonstrate that the technique produces correct numerical solution without spurious modes.

A bridging transition technique for the combination of meshfree method with finite element method in 2D solids and structures

Abstract: For certain continuum problems, it is desirable and beneficial to combine two different methods together in order to exploit their advantages while evading their disadvantages. In this paper, a bridging transition algorithm is developed for the combination of the meshfree method (MM) with the finite element method (FEM). In this coupled method, the MM is used in the sub-domain where the MM is required to obtain high accuracy, and the FEM is employed in other sub-domains where FEM is required to improve the computational efficiency. The MM domain and the FEM domain are connected by a transition (bridging) region. A modified variational formulation and the Lagrange multiplier method are used to ensure the compatibility of displacements and their gradients. To improve the computational efficiency and reduce the meshing cost in the transition region, regularly distributed transition particles, which are independent of either the meshfree nodes or the FE nodes, can be inserted into the transition region. The newly developed coupled method is applied to the stress analysis of 2D solids and structures in order to investigate its’ performance and study parameters. Numerical results show that the present coupled method is convergent, accurate and stable. The coupled method has a promising potential for practical applications, because it can take advantages of both the MM and FEM when overcome their shortcomings.

The connection between the finite difference like methods and the methods based on initial value problems for ode

Abstract: Publisher Summary This chapter highlights the connection between the finite difference like methods and the methods based on initial value problems forordinary differential equations (ODE). The formulation of the systems of equations stemming from different versions of finite difference methods or finite element methods, as presented in the theory, is not the entire story of the numerical solution of boundary value problems for ODE. An essential part of the process is the solution of the derived systems of equations. There are many ways to solve these systems, a variety of direct and iterative methods. Using the finite difference method, the initial value problem for equations can be solved by a second order method without automatic step selection. The use of the deferred corrections leads to the solution of the same equations using an extrapolation procedure based on second order schemes. Another point is that the second order method stemming from the elimination method needs a relatively very small number of operations. Iterative procedures are not used when solving boundary value problems for ODEs.

Adaptive meshless methods in electromagnetic modeling: A gradient-based refinement strategy

Abstract: Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic simulations in particular, this distinguishing characteristic translates into an enhanced capability for conformal and multi-scale modeling. The method also facilitates adaptive discretization refinements, which provides an important tool to decrease memory consumption and computation time. In this paper, a refinement strategy is introduced for RPIM. In the proposed node adaptation algorithm, the accuracy of a solution is increased iteratively based on an initial solution with a coarse discretization. In contrast to the commonly used residual-based adaptivity algorithms, this definition is extended by an error estimator based on the solution gradient. In the studied cases this strategy leads to increased convergence rates compared with the standard algorithm. Numerical examples are provided to illustrate the effectiveness of the algorithm.