Meshfree methods

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

An improved truly meshless method based on a new shape function and nodal integration

Abstract: An improved truly meshless method is presented for three-dimensional (3D) electromagnetic problems. In the proposed method, the computational time for the construction of the introduced shape function is lower than the other meshless methods considerably. An efficient and stable nodal integration technique based on the Taylor series extension is also used in the proposed meshless method. Weak-form formulations adopted for creating discretized system equations of electrostatic and electromagnetic 3D problems are also presented. In the proposed fast truly meshless method, unlike in traditional meshless schemes where background mesh is utilized to compute integrals, nodal integration is used to avoid meshing. The numerical solutions for electrostatic and electromagnetic problems show that the presented method is a robust meshfree method and possesses better computational properties compared with traditional meshless methods. Copyright © 2012 John Wiley & Sons, Ltd.

The quasi-uniformity condition for reproducing kernel element method meshes

Abstract: The reproducing kernel element method is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-δ property. To achieve these properties, the underlying mesh must meet certain regularity constraints. This paper develops a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. The algorithm is demonstrated on several mesh types. Finally, a guide to generation of quasi-uniform meshes is discussed.

Meshless methods for shear-deformable beams and plates based on mixed weak forms

Abstract: Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin plate theories have seen wide use throughout engineering practice to simulate the response of structures with planar dimensions far larger than their thickness dimension. Meshless methods have been applied to construct numerical methods to solve the shear deformable theories. Similarly to the finite element method, meshless methods must be carefully designed to over- come the well-known shear-locking problem. Many successful treatments of shear-locking in the finite element literature are constructed through the application of a mixed weak form. In the mixed weak form the shear stresses are treated as an independent variational quantity in addition to the usual displacement variables. We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam problem that converges to the stable First-order/zero-order finite element method in the local limit when using maximum entropy meshless basis functions. The resulting formulation is free from the effects shear-locking. We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking. Finally we consider the construction of a generalised displacement method where the shear stresses are eliminated prior to the solution of the final linear system of equations. We implement an existing technique in the literature for the Stokes problem called the nodal volume averaging technique. To ensure stability we split the shear energy between a part calculated using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from the effects of shear-locking.