Search Results (4)

This paper uses the flexural vibration of cantilever beams as a benchmark problem to test mesh-free and finite element methods in structural dynamics. First, a symbolic analysis of the “kernel collocation” type mesh-free method is carried out, in which the collocation function satisfies the boundary conditions. This enables both Finite Element (FE) and mesh-free results to be compared with exact analytical ones. Thereafter, the natural frequencies and Frequency Response Function (FRF), in terms of the beam parameters, are determined and compared with the analytical results, that exist in the literature. It is shown that by adjusting the parameters of the kernel function, we can find identical peaks to those of the analytical method. The finite element method is also employed to solve this problem, and the first three natural frequencies were computed in terms of the beam parameters. When comparing the two methods, we see that by increasing the number of elements in the FEM we can always achieve better accuracy, but we will obtain twice the number of modal frequencies. However, the mesh-free method with the same number of nodes does not provide these extra frequencies. From this benchmark problem, it is concluded that the accuracy of the mesh-free methods always depends on the adjustment of the kernel function. However, the FEM is advantageous because it does not require such adjustments. Full article

The accuracy of the conventional finite element (FE) approximation for the analysis of acoustic propagation is always characterized by an intractable numerical dispersion error. With the aim of enhancing the performance of the FE approximation for acoustics, a coupled FE-Meshfree numerical method based on triangular elements is proposed in this work. In the proposed new triangular element, the required local numerical approximation is built using point interpolation mesh-free techniques with polynomial-radial basis functions, and the original linear shape functions from the classical FE approximation are employed to satisfy the condition of partition of unity. Consequently, this coupled FE-Meshfree numerical method possesses simultaneously the strengths of the conventional FE approximation and the meshfree numerical techniques. From a number of representative numerical experiments of acoustic propagation, it is shown that in acoustic analysis, better numerical performance can be achieved by suppressing the numerical dispersion error by the proposed FE-Meshfree approximation in comparison with the FE approximation. More importantly, it also shows better numerical features in terms of convergence rate and computational efficiency than the original FE approach; hence, it is a very good alternative numerical approach to the existing methods in computational acoustics fields. Full article

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration. Full article

The simulation of large deformation problems, involving complex history-dependent constitutive laws, is of paramount importance in several engineering fields. Particular attention has to be paid to the choice of a suitable numerical technique such that reliable results can be obtained. In this paper, a Material Point Method (MPM) and a Galerkin Meshfree Method (GMM) are presented and verified against classical benchmarks in solid mechanics. The aim is to demonstrate the good behavior of the methods in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. The vast majority of MPM techniques in the literatrue are based on some sort of explicit time integration. The techniques proposed in the current work, on the contrary, are based on implicit approaches, which can also be easily adapted to the simulation of static cases. The two methods are presented so as to highlight the similarities to rather than the differences from “standard” Updated Lagrangian (UL) approaches commonly employed by the Finite Elements (FE) community. Although both methods are able to give a good prediction, it is observed that, under very large deformation of the medium, GMM lacks robustness due to its meshfree natrue, which makes the definition of the meshless shape functions more difficult and expensive than in MPM. On the other hand, the mesh-based MPM is demonstrated to be more robust and reliable for extremely large deformation cases. Full article