Meshfree methods

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

A Localized Radial-Basis-Function Meshless Method Approach to Axisymmetric Thermo-Elasticity

Abstract: Current methods for solving thermoelasticity problems involve using finite element analysis, boundary element analysis, or other meshed-type methods to determine the displacements under an imposed temperature/stress field. This paper will detail a new approach using localized meshless methods based on multi-quadric radial basis function interpolation to solve these types of coupled thermoelasticity problems. Here, a point distribution is used along with a localized collocation method to solve the Navier equation for the components of the displacement vector. The specific application considered in this paper is that of axisymmetric thermo-elasticity. With rapidly increasing availability and performance of computer workstations and clusters, the major time requirement for solving a thermoelasticity model is no longer the computation time, but rather the problem setup. Defining the required mesh for a complex geometry can be extremely complicated and time consuming, and new methods are desired that can reduce this time. The proposed meshless method features the complete elimination of a mesh, be it structured or unstructured, and the associated complexities involved in its generation and control. The reduction of initial model setup time makes the meshless approach an ideal method of solving coupled thermoelasticity problems. Several examples with exact solutions are used to verify this method for various geometries and boundary condition combinations.

Improved boundary tracking in meshless simulations of free-surface flows

Abstract: In this paper we review the use of shape constructors, particularly α-shapes, for the simulation of free-surface flow problems. This technique, in conjunction with meshless methods, allows for the simulation of such problems in an updated Lagrangian approach without the need for an explicit description of the boundary of the domain. At each time step, the shape of the domain is extracted automatically. However, it is well know that α-shape techniques present some drawbacks. The first is the choice of the α parameter, related to the level of detail to which the domain is represented. Also contact detection of free surfaces (auto-contact) or between the free surface and a rigid boundary, for instance, is often detected with an error of the order \({\mathcal{O}(h)}\) , the nodal spacing parameter, in the gap distance. We propose an heuristic technique for the choice of the α parameter and develop a novel methodology for an improved detection of contact or merging flows. The proposed technique is illustrated with the help of some examples in solid and fluid mechanics.

On preconditioned BiCGSTAB solver for MLPG method applied to heat conduction in 3D complex geometry

Abstract: Development of meshfree methods is in full swing in the last few decades as these methods overcome the shortcomings of mesh based methods. Development of meshfree methods for large scale problems requires fast iterative solvers. Preconditioner plays an important role in improving convergence rate of an iterative solver. Recently proposed SRJ method can be used as a stand-alone solver and as a preconditioner in CG type methods. Objective of the present work is to evaluate the performance of SRJ method as a preconditioner in BiCGSTAB method in connection with the MLPG method. Three dimensional Poisson equation and heat conduction equation are solved in regular and irregular domain respectively. The SRJ method is tested as a solver and as a preconditioner in BiCGSTAB method. Its performance as a preconditioner is compared with Jacobi and SOR preconditioners.

An ALE based hybrid meshfree local RBF-cartesian FD scheme for incompressible flow around moving boundaries

Abstract: A solution scheme is presented to simulate incompressible viscous flow around moving boundaries using hybrid meshfree-Cartesian grid. The presented solution approach avoids intensive re-meshing and enhances computational efficiency by combining the advantages of both meshfree and mesh-based methods for flow around moving objects. The scheme employs a body conformal meshfree nodal cloud around the solid object which convects with the moving solid boundary. On the outer side, meshfree nodal cloud is surrounded and partially overlapped by a stationary Cartesian grid. Navier Strokes equations in Arbitrary-Lagrangian-Eulerian (ALE) formulations are solved over moving nodal cloud using meshfree local Radial Basis Functions in finite difference Mode (RBF-FD). Eulerian form of flow equations are solved over static Cartesian grid using conventional finite difference scheme. Meshfree nodes can efficiently adapt to the moving boundary without necessitating re-meshing. Use of finite difference method over Cartesian grid allows faster computing and improves computational efficiency. Variation in computation time has been studied with corresponding change in size of meshfree and Cartesian grids. Significant reduction in computation time is achieved by reducing the size of meshfree cloud. The solution scheme is validated by simulating two dimensional flows around vibrating cylindrical objects. For this purpose, forced as well as vortex induced cylindrical vibration cases are investigated and solutions are compared with computational and experimental results available in literature.

Shape optimization of piezoelectric devices using an enriched meshfree method

Abstract: We present an enriched reproducing kernel particle method for shape sensitivity analysis and shape optimization of two-dimensional electromechanical domains. This meshfree method incorporates enrichment functions for better representation of discontinuous electromechanical fields across internal boundaries. We use cubic splines for delineating the geometry of internal/external domain boundaries; and the nodal coordinates and slopes of these splines at their control points become the design parameters. This approach enables smooth manipulations of bi-material interfaces and external boundaries during the optimization process. It also enables the calculation of displacement and electric-potential field sensitivities with respect to the design parameters through direct differentiation, for which we adopt the classical material derivative approach. We verify this implementation of sensitivity calculations against an exact solution to a variant of Lame's problem, and also, finite-difference approximations. We follow a sequential quadratic programming approach to minimize the cost function; and demonstrate the utility of the overall technique through a model problem that involves the shape optimization of a piezoelectric fan. Copyright © 2008 John Wiley & Sons, Ltd.