Meshfree methods

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

An isogeometric-meshfree coupling approach for analysis of cracks

Abstract: Summary This paper presents a new concurrent simulation approach to couple isogeometric analysis (IGA) with the meshfree method for studying of crack problems. In the present method, the overall physical domain is divided into two sub-domains which are formulated with the IGA and meshfree method, respectively. In the meshfree sub-domain, the moving least squares (MLS) shape function is adopted for the discretization of the area around crack tips, and the IGA sub-domain is adopted in the remaining area. Meanwhile, the interface region between the two sub-domains is represented by coupled shape functions. The resulting shape function, which comprises both IGA and meshfree shape functions, satisfies the consistency condition, thus ensuring convergence of the method. Moreover, the meshfree shape functions augmented with the enriched basis functions to predict the singular stress fields near a crack tip are presented. The proposed approach is also applied to simulate the crack propagation under a mixed-mode condition. Several numerical examples are studied to demonstrate the utility and robustness of the proposed method. This article is protected by copyright. All rights reserved.

Numerical Investigation of the Time Fractional Mobile-Immobile Advection-Dispersion Model Arising from Solute Transport in Porous Media

Abstract: Evolution equations containing fractional derivatives can offer efficient mathematical models for determination of anomalous diffusion and transport dynamics in multi-faceted systems that cannot be precisely modeled by using normal integer order equations. In recent times, researches have found out that lots of physical processes illustrate fractional order characteristics that alters with time or space. The continuum of order in the fractional calculus permits the order of the fractional operator be accounted for as a variable. In the current research work, radial basis functions (RBFs) approximation is utilized for solving fractional mobile-immobile advection-dispersion (TF-MIM-AD) model in a bounded domain which is applied for explaining solute transport in both porous and fractured media. In this approach, firstly, the discretization process of the aforesaid equation with of convergence order $$\mathcal {O}(\delta t^{})$$ in the t-direction is described via the finite difference scheme for $$ 0< \alpha <1$$ . Afterwards, by help of the meshless methods based on RBFs, we will illustrate how to obtain the approximated solution. The stability and convergence of time-discretized scheme are also theoretically discussed in detail throughout the paper. Finally, two numerical instances are included to clarify effectiveness and accuracy of our proposed concepts which is investigated in the current research work.

Local Maximum-Entropy Approximation Schemes

Abstract: We present a new approach to construct approximation schemes from scattered data on a node set, i.e. in the spirit of meshfree methods. The rational procedure behind these methods is to harmonize the locality of the shape functions and the information-theoretical optimality (entropy maximization) of the scheme, in a sense to be made precise in the paper. As a result, a one-parameter family of methods is defined, which smoothly and seamlessly bridges meshfree-style approximants and Delaunay approximants. Besides an appealing theoretical foundation, the method presents a number of practical advantages when it comes to solving partial differential equations. The non-negativity introduces the well-known monotonicity and variation-diminishing properties of the approximation scheme. Also, these methods satisfy ab initio a weak version of the Kronecker-delta property, which makes essential boundary conditions straightforward. The calculation of the shape functions is both efficient and robust in any spacial dimension. The implementation of a Galerkin method based on local maximum entropy approximants is illustrated by examples.