Meshfree methods

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

Two meshless methods for solving nonlinear ordinary differential equations in engineering and applied sciences

Abstract: Abstract In this paper, two meshless methods have been introduced to solve some nonlinear problems arising in engineering and applied sciences. These two methods include the operational matrix Bernstein polynomials and the operational matrix with Chebyshev polynomials. They provide an approximate solution by converting the nonlinear differential equation into a system of nonlinear algebraic equations, which is solved by using Mathematica® 10. Four applications, which are the well-known nonlinear problems: the magnetohydrodynamic squeezing fluid, the Jeffery-Hamel flow, the straight fin problem and the Falkner-Skan equation are presented and solved using the proposed methods. To illustrate the accuracy and efficiency of the proposed methods, the maximum error remainder is calculated. The results shown that the proposed methods are accurate, reliable, time saving and effective. In addition, the approximate solutions are compared with the fourth order Runge-Kutta method (RK4) achieving good agreements.

Local radial basis function meshless scheme for vector radiative transfer in participating media with randomly oriented axisymmetric particles.

Abstract: A local radial basis function meshless scheme (LRBFM) is developed to solve polarized radiative transfer in participating media containing randomly oriented axisymmetric particles in which radial basis functions augmented with polynomial basis are employed to construct the trial functions, and the vector radiative-transfer equation based on the discrete-ordinates approach is discretized directly by collocation method. The LRBFM belongs to a class of truly meshless methods that do not need any mesh or any numerical integration scheme. Performances of the LRBFM are verified with analytical solutions and other numerical results reported earlier in the literature via five various test cases. The predicted angular distribution of brightness temperature and Stokes vector by the LRBFM agree very well with the benchmark. It is demonstrated that the LRBFM is accurate to solve vector radiative transfer in participating media with randomly oriented axisymmetric particles.

A Petrov–Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations

Abstract: Meshfree methods with arbitrary order smooth approximation are very attractive for accurate numerical modeling of fractional differential equations, especially for multi-dimensional problems. However, the non-local property of fractional derivatives poses considerable difficulty and complexity for the numerical simulations of fractional differential equations and this issue becomes much more severe for meshfree methods due to the rational nature of their shape functions. In order to resolve this issue, a new weak formulation regarding multi-dimensional Riemann–Liouville fractional diffusion equations is introduced through unequally splitting the original fractional derivative of the governing equation into a fractional derivative for the weight function and an integer derivative for the trial function. Accordingly, a Petrov–Galerkin finite element-meshfree method is developed, where smooth reproducing kernel meshfree shape functions are adopted for the trial function approximation to enhance the solution accuracy, and the discretization of weight function is realized by the explicit finite element shape functions with an analytical fractional derivative evaluation to further reduce the computational complexity and improve efficiency. The proposed method enables a direct and efficient employment of meshfree approximation, and also eliminates the undesirable singular integration problem arising in the fractional derivative computation of meshfree shape functions. A nonlinear extension of the proposed method to the fractional Allen–Cahn equation is presented as well. The effectiveness of the proposed methodology is consistently demonstrated by numerical results.

A Local Meshless Method for Steady State Convection Dominated Flows

Abstract: A stable localized meshless method (SLMM) is proposed for convection dominated PDEs exhibiting boundary layer. Some cases of continuous and discontinuous boundary data as well as continuous and discontinuous source function with constant and variable convection coefficients are considered. In this approach, the localized meshless method is implemented on specialized sub-domains embedded with flow direction of underlying fluid. The proposed method is based on flow featured overlapping sub-domains, called stencils. The concept of flow direction is used to construct good quality stencils having the ability to capture flow features, such as boundary layer, accurately. Numerical experiments are presented to compare the proposed method with the finite-difference method on special grid (FDSG), the standard finite-element method, hybridized SUPG method, hybridized upwind method, residual-free bubbles (RFB) method and other meshless methods. Numerical results confirm that the new approach is accurate and efficient for solving a wide class of one-, two-, and three-dimensional convection-dominated PDEs. In some cases, performance of the SLMM is comparable and sometimes better than the mesh-based finite-element and finite-difference methods.

Meshfree analysis of functionally graded plates with a novel four-unknown arctangent exponential shear deformation theory

Abstract: A novel refined arctangent exponential shear deformation theory (RAESDT) is presented for analysis the mechanical behavior of both isotropic and sandwich FGM plates. Material properties are set to ...