H-version Of Finite Element Method Research Articles

Identification of Cracks in Functionally Graded Material Beams Using the H-Version of Finite Element Method

In this article, we analyze the effect of transverse cracks on the natural frequencies of a Euler-Bernoulli functional gradient beam. The studied beam was discretized into finite elements and the global matrices of the motion equation are determined by applying the Lagrange equation to the beam kinetic and deformation energies. The material properties are considered to vary in the directions of the beam thickness, the gradation is described by the power-law distribution, the stiffness of the cracked element is determined based on the reduction of the beam cross-section. The numerical results obtained are compared with those available in the previous study. Finally, case studies were carried out to analyses the influence of the power law index, the depth and the opposition of the crack on the natural frequencies of the beam for different boundary conditions; these studies demonstrate the advantage of the FGM beam over the purely metal beam.

An enriched finite element method for efficient solutions of transient heat diffusion problems with multiple heat sources

We propose an efficient formulation using the framework of Generalized Finite Element Method (GFEM) for the solutions of transient heat diffusion problems having multiple heat sources in the solution domain. The purpose here is to use minimal computational resources and rely on coarse mesh grids to capture the sharp variations of temperature field. We use Gaussian functions of global nature to enrich the GFEM approximation space which ensure efficient solution in the whole solution domain. To capture steep thermal gradients at multiple locations, a multiplicity of enrichment functions is used and the peaks of enrichment functions are centred at the cores of heat sources. The advantage of this approach is that no further degrees of freedom (DOFs) are added to capture the solution at multiple locations. Besides, the enrichment functions are time-independent; the temporal variation of temperature is embedded in the definition of the enrichment functions. This formulation requires the assembly of system matrix once and only the right-hand side of the system of equations is updated at subsequent time steps which results in a significant reduction in the overall computational time. We consider problems in two-dimensional (2D) and three-dimensional (3D) domains to show the effectiveness of the proposed approach. A reduction of more than 95% in DOFs and more than 80% in computational time is achieved as compared to the h-version of finite element method.

An adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses

In this paper, a first application of an adaptive Generalized Finite Element Method to free longitudinal vibration analysis of straight bars and trusses is presented. The Generalized Finite Element Method is developed by enriching the standard Finite Element Method space, whose basis performs a partition of unity, with knowledge about the differential equation being solved. The enrichment functions used are dependent on the geometric and mechanical properties of the element. The proposed approach converges very fast and is able to approximate the frequency related to any vibration mode. The variational problem of free vibration is formulated and the main aspects of the adaptive Generalized Finite Element Method are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform straight bars are checked. The application of this technique in a truss is also presented. The frequencies obtained by the adaptive Generalized Finite Element Method are compared with those obtained by the analytical solution, the Composite Element Method and the h-version of Finite Element Method.