Meshfree methods

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

H-matrix preconditioners for saddle-point systems from meshfree discretization

Abstract: In this paper we describe and compare preconditioners for saddle-point systems obtained from meshfree discretizations, using the concepts of hierarchical (or H-)matrices. Previous work by the authors using this approach did not use H-matrix techniques throughout, as is done here. Comparison shows the method described here to be better than the author’s previous method, an AMG method adapted to saddle point systems, and conventional iterative methods such as JOR.

Investigation on meshfree particle methods for fluid structure interaction problems

Abstract: The research aims to investigate the application of meshfree particle methods for computational modelling of the Fluid Structure Interaction problems with particular emphasis on flow around cylindrical objects and aerofoils. For this purpose, a solution scheme has been developed for solving incompressible, viscous Navier-Stokes (N-S) equations over meshfree particles. Spatial derivatives appearing in N-S equations are dealt with using radial basis functions infinite difference mode (RBF-FD). A comparative study has also been conducted between implicit and explicit in time solution schemes for N-S equations over meshfree nodes. Subsequently, a coupled meshfree and mesh-based solution scheme is proposed, over hybrid fluid grid, for incompressible, viscous flow around stationary as well as moving objects. The aim of this coupled solver is to provide efficiency and flexibility by combining the advantages of both meshfree and mesh-based methods. The coupled solution scheme suggests generating a body conformal meshfree nodal cloud around the solid body in the near field. A static Cartesian grid surrounds the meshfree cloud in the far field. The Meshfree nodes offer flexibility in dealing with solid motion by moving along the solid boundary without necessitating re-meshing. The Cartesian grid, on the other hand, provides improved performance by allowing faster computation owing to the use of efficient mesh based method. Flow equations, in Arbitrary Lagrangian-Eulerian (ALE) formulation, are solved using RBF-FD based scheme over moving meshfree nodes. Conventional infinite differencing is used over static Cartesian grid for flow equations in Eulerian formulation. The coupled solution scheme, on hybrid grid, is employed for closely coupled Fluid Structure Interaction problems. The equations for solid motion are solved using classical Runge-Kutta method. Close coupling between fluid and structural solvers is realized by a sub-iterative prediction-correction algorithm. In order to reduce computational overhead due to sub-iterations, only near field flow (in meshfree zone) is solved during inner iterations. Solution over full fluid domain is sought during outer (time step) iterations only, when the convergence at fluidsolid interface has already been reached. The solution scheme is also applied for high Reynolds number problems. For this purpose, a stabilization term is included in the flow equations to suppress the spurious oscillations. The stabilization term is derived using vimomentum balance equation over control volume and applying higher order Taylor series expansion of momentum flux and fluid forces. In order to avoid ill-conditioning and accuracy problems related to RBF matrices in domains having varying nodal density, use of shape adaptive RBFs are proposed. In that, the shape parameter of the radial basis function is varied according to local nodal density. Moreover, adaptive sizing of influence domain has also been introduced to maintain suitable number of neighbouring particles. These adaptive techniques are found to be useful as they allow much finer nodal distribution at regions of interest enabling accurate capturing of flow gradients and leading to better results. The use of hybrid grid offers flexibility in dealing with moving boundaries. Moreover, in addition to allowing faster computing over Cartesian grid, it also enables using the reduced fluid domain during inner FSI iterations and therefore helps reduce the number of computations in the fluid domain during fluid-solid coupling. The solution scheme was tested for problems relating to flows around static as well as moving cylinders and aerofoils. Flow induced vibrations have been studied with one and two degrees of freedom. The results are found to be in good agreement with previous numerical work and experimental results.

Local multiquadric RBF meshless scheme for radiative heat transfer in strongly inhomogeneous media

Abstract: A local radial basis function meshless method (LRBFM) is developed to solve radiative heat transfer in participating media, in which multiquadric (MQ) radial basis functions (RBF) augmented with polynomial basis are employed to construct the trial functions, and the radiative transfer equation (RTE) is discretized directly at nodes by collocation method. The LRBFM belongs to a class of truly meshless methods which do not need any mesh, and can be implemented on a set of uniform or irregular nodes without nodes׳ connectivity. To improve numerical stability of LRBFM for the solution to radiative heat transfer in strongly inhomogeneous media, an upwind support domain scheme is introduced. The upwind scheme is implemented by moving the support domain of local radial basis function interpolation approximation to the opposite direction of each streamline, which can fully capture the information from upstream and improve the accuracy and stability of LRBFM. Performances of the LRBFM and upwind LRBFM (LRBFM_U) are compared with analytical solutions and other numerical results reported earlier in the literatures via a variety of problems in 1-D and 2-D geometries with strongly inhomogeneous media. It is demonstrated that the local radial basis function meshless method with upwind support domain scheme (LRBFM_U) provides high accuracy and great stability to solve radiative heat transfer in strongly inhomogeneous media.

Meshless analysis of shear deformable shells: boundary and interface constraints

Abstract: Meshless methods provide a highly continuous approximation field, convenient for thin structures like shells. Nevertheless, the lack of Kronecker Delta property makes the formulation of essential boundary conditions not straightforward, as the trial and test fields cannot be tailored to boundary values. Similar problem arise when different approximation regions must be joined, in a multi-region problem, such as kinks, folds or joints. This work presents three approaches to impose both kinematic conditions: the well-known Lagrange multiplier method, used since the beginning of the element free Galerkin method; a pure penalty approach; and the recently rediscovered alternative of Nitsche's method. We use the discretization technique for thick Reissner---Mindlin shells and adapt the weak form as to separate displacement and rotational degrees of freedom and obtain suitable and separate stabilization parameters. This approach enables the modeling of discontinuous shells and local refinement on multi-region problems.