Structural dynamic analysis for time response of bars and trusses using the generalized finite element method

PurposeThe purpose of this paper is to evaluate the capabilities of the generalized finite element method (GFEM) under the context of the geometrically nonlinear analysis. The effect of large displacements and deformations, typical of such analysis, induces a significant distortion of the element mesh, penalizing the quality of the standard finite element method approximation. The main concern here is to identify how the enrichment strategy from GFEM, that usually makes this method less susceptible to the mesh distortion, may be used under the total and updated Lagrangian formulations.Design/methodology/approachAn existing computational environment that allows linear and nonlinear analysis, has been used to implement the analysis with geometric nonlinearity by GFEM, using different polynomial enrichments.FindingsThe geometrically nonlinear analysis using total and updated Lagrangian formulations are considered in GFEM. Classical problems are numerically simulated and the accuracy and robustness of the GFEM are highlighted.Originality/valueThis study shows a novel study about GFEM analysis using a complete polynomial space to enrich the approximation of the geometrically nonlinear analysis adopting the total and updated Lagrangian formulations. This strategy guarantees the good precision of the analysis for higher level of mesh distortion in the case of the total Lagrangian formulation. On the other hand, in the updated Lagrangian approach, the need of updating the degrees of freedom during the incremental and iterative solution are for the first time identified and discussed here.

The Orthonormalized Generalized Finite Element Method–OGFEM: Efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions

Bridging Scales with a Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h l?+?k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.

This paper presents an analysis of interacting cracks using a generalized finite element method (GFEM) enriched with so-called global-local functions. In this approach, solutions of local boundary value problems computed in a global-local analysis are used to enrich the global approximation space through the partition of unity framework used in the GFEM. This approach is related to the global-local procedure in the FEM, which is broadly used in industry to analyze fracture mechanics problems in complex three-dimensional geometries. In this paper, we compare the effectiveness of the global-local FEM with the GFEM with global-local enrichment functions. Numerical experiments demonstrate that the latter is much more robust than the former. In particular, the GFEM is less sensitive to the quality of boundary conditions applied to local problems than the global-local FEM. Stress intensity factors computed with the conventional global-local approach showed errors of up to one order of magnitude larger than in the case of the GFEM. The numerical experiments also demonstrate that the GFEM can account for interactions among cracks with different scale sizes, even when not all cracks are modeled in the global domain.

Strongly stable generalized finite element method: Application to interface problems

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

The Generalized Finite Element Method (GFEM) is a numerical method based on the Finite Element Method (FEM), presenting as its main feature the possibility of improving the solution by means of local enrichment functions. In spite of its advantages, the method demands a complex data structure, which can be especially benefited by the Object-Oriented Programming (OOP). Even though the OOP for the traditional FEM has been extensively described in the technical literature, specific design issues related to the GFEM are yet little discussed and not clearly defined. In the present article it is described an Object-Oriented (OO) class design for the GFEM, aiming to achieve a computational code that presents a flexible class structure, circumventing the difficulties associated to the method characteristics. The proposed design is evaluated by means of some numerical examples, computed using a code implemented in Python programming language.

The main objectives of this paper are to simulate 3-D fatigue crack propagation using a Generalized Finite Element Method (GFEM) and to validate the results against experimental data. This GFEM adopts a high-order p-hierarchical basis and explicit representations of crack surfaces. Both h-refinement around the fracture fronts and non-uniform p-enrichment of the analysis domain are used to control discretization errors. A systematic validation of this GFEM applied to 3-D fatigue crack propagation has not been reported in the literature. The Displacement Correlation Method (DCM) is used to extract stress intensity factors. The effect of material parameters adopted in the DCM on the crack growth rate and fracture shape is investigated. Three increasingly complex fatigue crack propagation problems are solved. The first involves mixed-mode loading in a modified compact tension specimen. The second one involves the transition from 2-D to 1-D crack surfaces and interactions between the crack front and the corners of the domain boundary. The final problem simulates the growth of a circumferential surface crack in a steel pipe subjected to fatigue bending with overloading, where interactions between the crack and the pipe’s inner surface result in the splitting of the crack front. Another contribution is an algorithm designed to manage cyclic load histories featuring variable loading ranges and ratios between minimum and maximum load magnitudes.

To solve wave propagation problems over large domains composed by different media, many authors use the generalized finite element method (GFEM) with a geometrically conforming partition and plane wave enrichment.Such analysis is often accomplished by decomposing the entire domain into several subdomains and performing the analysis individually on each one. Then, the global analysis can be realized by using Lagrange multipliers to ensure the interface constraints. In this article, the GFEM with plane wave enrichment is extended to nonconforming discrete cases where the interface between the media can be handled as piecewise linear segments. Results for problems for which the analytical solution is known are presented to demonstrate the efficiency of the proposed technique. The convergence of the method is also presented as a function of the number of plane wave directions. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2709–2716, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.27166

Although simple finite elements as truss members and beam members in frames have been used in structural analysis for many years, the general finite element methods of analysis have been developped only recently in parallel with the development of digital computers. It is the use of general matrix methods in high speed digital computers that makes the general finite element methods feasible. At the present time the finite element methods are being used extensively in the analysis and design of flight vehicle structures as well as in other types of structures. Many large general purpose computer programs as well as many specialized programs have been developed for the analysis of structures. These computer programs use various finite elements and usually calculate and assemble the element matrices by one of the three methods of analysis described in previous chapters. Most of the general programs use the displacement method of analysis; a few general programs and many specialized programs use the force method of analysis; a few programs use the mixed method, while some programs use more than one method.

Coupled dynamic analysis for wave interaction with a truss spar and its mooring line /riser system in time domain

Finite-element-based techniques are one of the most popular methods used to model electromagnetic field behavior, but rely on the underlying tesselation to construct the ansatz space. Recently, Babuska and his colleagues developed the generalized finite-element method (GFEM), which overcomes this constraint and admits a larger class of basis functions. Application of this technique has been largely restricted to Poisson systems. In this paper, we explore the applicability of this technique to two-dimensional Helmholtz systems. We investigate methods necessary to impose various boundary conditions. We use this analysis to build the framework for hybridizing boundary integral techniques with GFEM, thus imposing an exact boundary condition to truncate the computational domain. We validate the results against analytical data for canonical geometries, and we demonstrate h and p convergence of this technique. Finally, to further validate the proposed approach for complex scatterers, we augment GFEM with perfectly matched layers, and compare it against the results obtained by using boundary integral GFEM

Nonlinear and dynamic structural analysis using combined approximations

PurposeThis paper aims to present an adaptive approach of the generalized finite element method (GFEM) for transient dynamic analysis of bars and trusses. The adaptive GFEM, previously proposed for free vibration analysis, is used with the modal superposition method to obtain precise time-history responses.Design/methodology/approachThe adaptive GFEM is applied to the transient analysis of bars and trusses. To increase the precision of the results and computational efficiency, the modal matrix is responsible for the decoupling of the dynamic equilibrium equations in the modal superposition method, which is used with only the presence of the problem’s most preponderant modes of vibration. These modes of vibration are identified by a proposed coefficient capable of indicating the influence of each mode on the transient response.FindingsThe proposed approach leads to more accurate results of displacement, velocity and acceleration when compared to the traditional finite element method.Originality/valueIn this paper, the application of the adaptive GFEM to the transient analysis of bars and trusses is presented for the first time. A methodology of identification of the preponderant modes to be retained in the modal matrix is proposed to improve the quality of the solution. The examples showed that the method has a strong potential to solve dynamic analysis problems, as the approach had already proved to be efficient in the modal analysis of different framed structures. A simple way to performh-refinement of truss elements to obtain reference solutions for dynamic problems is also proposed.