Meshfree methods

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

A meshfree radial basis function method for simulation of multi‐dimensional conservation problems

Abstract: Many computational fluid dynamics problems utilize finite volume frameworks for simulation, due to the simplifications provided by conservative formulation of the driving partial differential equations (PDEs). However, fluid dynamics applications can often involve temporal shifts in the domain structure—such as moving boundaries, or pore structure changes—requiring mesh adaptation throughout computation. These mesh adaptations often render classical numerical methods such as the finite volume method infeasible, due to their reliance on a well‐defined static mesh structure. This limitation has led to the development of a wide variety of meshless methods—techniques that can simulate PDEs without requiring a rigid connective structure between nodes. However, most meshless methods are typically based on finite element or finite difference formulations, and the limited number of meshless finite volume methods (MFVMs) either introduce a weak background mesh, or use weak‐form approximations that do not take full advantage of the strong conservative form of the driving equations. Addressing this gap within this study we outline a meshfree numerical scheme for simulation of partial different equations, based on strong‐form finite volume style formulations. Building upon the previously developed MFVM, this technique uses radial basis functions to interpolate the problem domain, and approximate fluxes in a disjoint finite volume scheme, removing reliance on a mesh structure. We present method derivation, including promising new techniques for enforcing boundary conditions in a meshless environment. Following this we discuss method accuracy and computational performance across a variety of problems in two and three dimensions. We then illustrate how this method may prove beneficial for applications in porous media modeling, and computational fluid dynamics. For completeness, we provide a sensitivity analysis of the method hyper‐parameters and investigate the conservative properties of the method. We also illustrate similarities of this approach to the widely used meshless point collocation methods. We close with a discussion of the strengths, limitations, and broader applicability of the technique.

Theoretical Approach to Element Free Galerkin Method and Its Mathematical Implementation

Abstract: Numerical methods such as FVM, FDM, FVM, and BVM are eminent for solving the physical problems in engineering and science. Mentioned numerical methods are based on the predefined topological map, generally called “mesh,” Meshes are required to establish the relations between nodes, which becomes vital for the creation of shape functions. The problems with mesh-based methods are (i) They require the qualitative mesh, which is a somewhat tedious, time-consuming & messy task (ii) Meshing & re-meshing for a sizeable computational domain is time consuming, tedious, and costly task also requires the skills (iii) In complex computational domains, the mesh-based method fails in terms of accuracy (iv) Glass hour and shear locking phenomena generally found in the traditional finite element method. In the last two and a half decades, many engineers and mathematicians have proposed a new class of numerical methods known as meshfree methods. Meshfree methods are independent of mesh and approximate the governing PDE based on the set of nodes only. This chapter seeds light on an eminent meshfree method called EFGM. Chapter deals with the introduction and background of meshfree methods, the EFGM method, and its mathematical formulations. The chapter also comprises two elastostatic numerical problems, the 1D problem of a bar with body forces and 2D Timoshenko cantilever beam with traction at the tip, numerical results have been evaluated & compared with exact results. The convergence of both 1D and 2D problems have been discussed. This work built a sound foundation on EFGM and will act as a stepping stone for novices in the field of meshfree methods. Keywords: Advanced numerical approach, Mesh-free methods, Element free Galerkin method

Efficient evaluation of stress intensity factors using a coupled BEM-SBFEM algorithm

Abstract: The scaled boundary finite element method (SBFEM) is gaining more recognition as an alternative to other element-based and meshless methods. Originally conceived as a tool for computing the dynamic stiffness of an unbounded domain [1], the method has since demonstrated greater versatility. One such area is its application to linear elastic stress analysis, particularly in fracture mechanics. The method works numerically in the circumferential direction making the usual piecewise polynomial approximation. However this is then coupled to an analytical solution in the radial direction. The method is especially useful in calculating stress intensity factors (SIFs) as they are functions of a displacement found semi-analytically rather than fully numerically [2]. The SBFEM is not without its drawbacks however. Its use in engineering domains containing holes and other similar features requires cumbersome substructuring [3]. The method also requires computationally expensive matrix manipulation, including matrix inversion and the solution of an eigenvalue system whose dimensions are twice the number of degrees of freedom of the model. The boundary element method (BEM) has also been developed to solve fracture mechanics problems. However, it makes only piecewise polynomial approximations to functions that are not polynomial in nature. The dual BEM allows efficient boundaryonly crack modelling [4], but requires the evaluation of hypersingular integrals and still uses piecewise polynomial shape functions. A coupled BEM-SBFEM system provides for the efficient calculation of SIFs for complex domains. A domain is subdivided into two regions. The crack tip is modelled by a small SBFEM subdomain with few degrees of freedom, thus keeping SBFEM computations to a minimum. This subdomain is coupled to a generally larger BEM subdomain that models the rest of the domain, thus exploiting the BEM’s ability to model geometric features efficiently.